94Surface microtopography of crystal faces

5.3 Spiral steps

An F face containing more than two PBCs grows either by two-dimensional nucleation or by spiral growth mediated by screw dislocations. Since the growth source is not fixed in layer growth through two-dimensional nucleation, and growth centers will fluctuate, appearing and disappearing during the growth process, it is not likely that a growth hillock with one summit will be formed by this mechanism. It is conjectured that island-like layers may be formed on terraced surfaces of layers of growth spirals, but the appearance of pyramidal growth hillocks with single summits will be exceptional. There is, however, the possibility of repeated nucleation on the same site, which may possibly result in the formation of pyramidal growth hillocks, but this model is unrealistic unless the mechanism of repeated nucleation is verified. When observing the growth process of a dislocation-free crystal face by an interferometric technique in situ, it was observed that growth centers appear, disappear, and reappear at different sites [10]. Although island-like steps on terraced surfaces may sometimes be observed by AFM, step patterns resembling contour lines on a geographic map (which are exclusively spiral growth steps originating from screw dislocations) are commonly observed on low-index F faces, which develop to determine the Habitus.

Within spiral growth steps, there exist the following types: elemental spirals with step heights equal to the unit cell or of monomolecular size, originating from isolated single screw dislocations; composite spirals spreading from screw dislocations; macro-steps originating from screw dislocations with large Burgers vector; and macro-steps formed by the bunching of elemental steps during their advancement.

Features characterizing these steps include

(1)whether an elemental spiral is circular or polygonal, and, if it is polygonal, its symmetry;

(2)the separation between the neighboring steps;

(3)the morphological changes that occur when elemental steps bunch together to form macro-steps.

These three features correspond to the two-dimensional morphology of a crystal, and are directly related to the problems of the three-dimensional morphology of polyhedral crystals, Habitus and Tracht. This is because the normal growth rate R which determines Habitus and Tracht is related in the following way to the height of a step, h, the advancing rate of the step, v, and the step separation, 0:

R hv/ 0

(see Fig. 5.4). The height of an elemental spiral step is determined by the Burgers vector of the screw dislocation. Since this corresponds to the size of a unit cell or

5.4 Circular and polygonal spirals 95

Figure 5.4. Relationship between normal growth rate R of a crystal face and the step

height h, the advancing rate v, and the step separation 0 of a growth spiral.

monomolecule, the height depends on the crystal face. In general, h becomes smaller as the morphological importance increases (the Bravais empirical rule; see Section 4.2).

Growth spirals may also sometimes be formed from screw dislocations with a large Burgers vector, but since screw dislocations are energetically disadvantageous, they tend to dissociate into elementary steps. In crystals having zigzag stacking in the structure, spiral steps originating from a dislocation with a Burgers vector of unit cell height dissociate into two or three spiral steps with opposite orientations, forming interlaced spiral patterns, which will be described in Section 5.5.

5.4Circular and polygonal spirals

The steps of spiral growth layers are one-dimensional interfaces. If the step is rough, the advancing rate is isotropic, forming a circular step pattern. The spiral pattern will be an Archimedean-type spiral. If a step is smooth, the spiral will be polygonal. Since the roughness of a step is controlled by strong bonds in the plane, i.e. PBCs, in addition to the growth parameters, the polygonal form will follow the symmetry elements involved in the crystal face. So, a square form with a four-fold symmetry axis will be expected on a {100} face of a crystal belonging to the cubic system m3m; a triangular form with a three-fold symmetry axis will be expected on the {111} face of a crystal belonging to the cubic system; a hexagonal form will appear on the {0001} face of a hexagonal system; a triangular form with a threefold symmetry axis will be expected on the {0001} face of a trigonal system; and a polygonal spiral containing a symmetry plane only on the {001} face of a monoclinic system or on the {1011} faces of hexagonal and trigonal systems. A few examples of polygonal growth spirals are shown in Fig. 5.5.

Since the one-dimensional roughness of the steps determines whether a spiral takes circular or polygonal form, these morphologies may be treated similarly to the roughening transition of an interface, as described in Chapter 3. It is possible to predict interface roughness either by Jackson’s factor or by Bennema–Gilmer’s generalized G factor (see Section 3.8). The coefficients which determine the

96 Surface microtopography of crystal faces

(c)

Figure 5.5. Examples of polygonal spirals. (a) SiC, 6H polytype, (0001). (b) Phlogopite,

1M polytype, (001). (c) SiC, 15R polytype, (0001).

factor are the orientational factor , the latent heat L, the melting point TM (in Jackson’s formula), the bonding energy in the solid and liquid, ss and ff , respectively, the solute–solvent interaction energy sf , and the growth temperature TG (in Bennema–Gilmer’s formula). The morphology of the elemental spirals will be modified depending on the factors that influence these coefficients. It is anticipated that polygonal spirals will be seen on crystal faces with a higher order of morphological importance, and that more rounded spirals will be observed as the order of morphological importance lowers. On hexagonal prismatic crystals of

5.4 Circular and polygonal spirals 97

(b)

(a)

(c)

Figure 5.6. Differential interference contrast photomicrographs showing the

¯ ¯

morphology of growth spirals observed on (a) (0001), (b) (2131), and (c) (1010) faces of synthetic emerald.

hydrothermally synthesized emeralds, the {0001} face shows spirals with regular hexagonal form, rectangular spirals with rounded corners on {1010}, and a spindle-like form on {2131} (see Fig. 5.6) [6].

On quartz crystals synthesized in a hydrothermal solution of NaOH, KOH series, {1010} faces show polygonal spiral steps, whereas rounded step patterns are observed on {1011} and {0111}.

The effect of factors which affect the factor, such as growth temperature or solute–solvent interaction, may be observed by comparing the surface micro-