98Surface microtopography of crystal faces

topographs of the same face on the same crystal species grown under different conditions. If we compare the spiral steps observed on the {1011} faces of natural and synthetic quartz, or among natural crystals formed under different growth conditions, we see these differences in growth conditions clearly. Polygonized step patterns are generally observed on {1011} faces of natural quartz, whereas circular patterns are commonly observed on the same faces of synthetic quartz.

5.5 Interlaced patterns

Crystals having a zigzag stacking in their structure can take different crystal systems or structures due to kinetic reasons, even if they grow under the same thermodynamic conditions. This phenomenon is called polytypism, and is distinguishable from polymorphism. The phenomenon has been observed widely among crystals with a layer structure, such as SiC, CdI2, and mica [11]. Polytypes are represented by the number of stacking layers followed by the letter denoting the crystal system. For example, 4H, 6H polytypes of SiC are polytypes with unit cells of a hexagonal (H) system, consisting of four and six layers, respectively; and 15R denotes a polytype consisting of three layers of a 32 stacking sequence ((3 2) 3 15 layers altogether), forming a rhombohedral (R) unit cell. In mica, polytypes of 1M, 2M1, 2M2, 3T, and higher orders, are known.

In this layered type of structure, for instance the upper two and lower two layers in the structure of the 4H polytype of SiC, the stacking sequence is reversed. Since it belongs to a hexagonal system, we expect a growth spiral with hexagonal form on the (0001) face, but the steps of the upper and lower layers advance as two oppositely oriented triangular layers, and so the upper layer, with the higher advancing rate, will unite with the lower layer to form single steps with unit cell height in the six edge directions. In the six directions along the corners of hexagonal form, however, two layers will never unite. As a result, an interlaced pattern appears in these directions. A schematic and actual examples of interlaced spirals observed on magnetoplumbite and SiC 6H polytype are shown in Fig. 5.7. In the mica polytypes 2M, 2O, and 3T, the elemental form of a single layer is five-sided, i.e. it is a truncated hexagon, and stacking occurs by rotation of the elemental layer by 60°, 180°, and 120°, and the resultant interlaced patterns become more complicated than those of SiC. Since the patterns of the steps of elemental growth spiral reflect very well the characteristics of the structure, it is possible to identify polytypes based on observations of these steps. Since the advancement of the steps is isotropic for circular spirals, the interlaced patterns will not appear unless the isotropy is violated.

In crystals containing no zigzag stacking, the orientations of the polygonal elemental spiral steps may be used as a criterion to identify whether the crystal is twinned or contains stacking faults.

5.5 Interlaced patterns 99

(a)

(b)

(c)

Figure 5.7. Phase contrast photomicrographs depicting interlaced spiral steps observed on (0001) faces of (a) magnetoplumbite and (b) SiC 6H polytype. (c) Schematic figure. The very narrow step separations observed at the center of the spiral in (b) are due to a sharp increment of supersaturation at the final stage due to the discontinued electrical supply.

100Surface microtopography of crystal faces

5.6 Step separation

Elemental growth spiral layers originating from an isolated dislocation can advance, keeping the step separation constant, unless factors which affect the advancing rate of the spiral steps, such as a local fluctuation in driving force or impurity adsorption, takes place. The step separation of a spiral, 0, is related to the critical radius of two-dimensional nuclei, rc, in the following manner (see ref. [11], Chapter 3):

0 19rc ,

and rc has the following relation to step free energy and the driving force:

G (rc) 16 3(v)2/(3 2),

where v is the molecular volume of the bulk nucleated phase. Therefore, 0 becomes smaller as the driving force increases. Also, 0 will vary with changes in reflecting the difference in temperatures, impurities, and solute–solvent interaction, and this results in the variation in the normal growth rates of crystal faces. Therefore, the factors affecting the Tracht and Habitus of polyhedral crystals are the same as those that affect the roughness of the steps and 0.

When the step separation is wide enough, typical spiral step patterns observable by optical microscopy may appear, but if the separation becomes narrower than the resolution power of the optical microscope, the spirals appear in the forms of polygonal pyramids or conical growth hillocks. Even if spiral patterns are not directly observable, we may assume that these growth hillocks are formed by the spiral growth mechanism. Examples representing the two cases are compared in Fig. 5.8.

Step separations vary depending on crystal faces, environmental phases, and the growth conditions. The step separation 0 on a crystal face of higher morphological importance is wider than that on a face of lower morphological importance. This may be compared with the difference observed for different faces of a crystal grown under the same growth conditions, polygonal or circular (Fig. 5.6). The 0 of growth spiral layers on the {0001} face of SiC or CdI2, or on the {001} face of mica, which have strong anisotropy in bonding, are wide and attain micrometer order. Since the step heights are of nanometer order, the profile of growth spiral hillocks on these faces attain a ratio of order 103 4 in step separation 0 and step height h. To get a feel for the scale of this, we could imagine walking for 1000 to 10 000 meters on an atomically flat surface before meeting a cliff one meter in height. It should be remembered that there is a big difference between the real profile and the images deduced from a schematic figure of spiral growth or optical photomicrographs of growth spirals.

The values of 0/h are different for different ambient phases and growth conditions. The ratio is of the order 103 4 for crystals grown from the vapor phase,