56Crystal growth

3.14 Element partitioning

When impurity atoms or ions are present in an ambient phase, they behave differently from the crystallizing component on arriving at the interfaces. To what extent these impurity components are incorporated into the growing crystal will differ depending on thermodynamic conditions, i.e. temperature and pressure, as well as on crystal chemical properties such as ionic size, and properties and spatial size of substitutional positions in the crystal, and also will be different depending on growth rate and crystallographic directions. Atoms and ions which are difficult to incorporate into a growing crystal will be removed from the crystal and will accumulate on the interface completely when the growth rate is slow, but at higher growth rates some will be enclosed by the growing crystal. Since the growth rate is dependent on the interface structure, and thus on the growth mechanism, the concentration of impurity components that are incorporated into the crystal will be different depending on crystallographic direction, namely their contents will be different depending on growth sectors (see Section 6.1).

To which side (ambient phase or crystal) and to what extent impurity components concentrate is called the “distribution (partition) of the element,” and the ratio is represented by the distribution (partition) coefficient. If the concentrations of the impurity component i in the ambient phase and in the crystal are expressed as cL and cS, respectively, the ratio cL/cS k is the distribution coefficient. Also called the “segregation coefficient,” k is determined by thermodynamic parameters such as temperature and pressure, but is not the same as the k calculated under equilibrium conditions, where the growth rate is zero (this k is expressed as k0). We may define the distribution coefficient during the growth as the effective distribution coefficient, keff . Impurity elements with keff 1 are those elements that are difficult to incorporate into the crystal in this ratio, and thus they sweep out to and concentrate at the interface; elements with keff 1 behave in the opposite manner. Elements with keff 1 that are swept out to the interface form a concentrated layer with a gradient that depends on the growth rate, R, if we assume a diffusion layer with thickness . Under the equilibrium condition where R 0, there is no gradient, but at rapid quenching, with R , the gradient is the steepest, and at a finite

R the gradient is intermediate. At R , impurity concentrations are equal in the ambient phase and in the crystal, and keff 1, whereas under equilibrium conditions, where R 0, keff k0. Assuming the diffusion constant of an impurity to be D, Burton, Prim, and Slichter [24] expressed keff by the following formula:

keff k0/[k0 (1 k0) exp ( R /D)].

This is the famous BPS equation; it shows that keff depends on R, but it does not show that it depends on orientation.

3.15 Inclusions 57

The first example which demonstrated that element partitioning has orientation dependence was the observation of the distribution of impurity Bi atoms in a Si single crystal grown from the melt. The Bi atoms are mainly distributed in the smooth interface growth sector. This was explained as being due to a much larger advancing rate of growth steps on a smooth interface than the normal growth rate

R of a rough interface. If the advancing rate is rapid, impurities arriving at the interface will be captured by the advancing growth steps before they are swept to the ambient phase. If we extend this concept further, it is predicted that impurity partitioning will depend on crystallographic orientation, and that the growth sectors of a face with the highest morphological importance will contain more impurities with keff 1. Impurity distribution in a single crystal cannot be homogeneous and will be dependent on growth sectors.

Impurity elements with keff 1 will concentrate on the interface. This affects the morphological stability of the interface through growth rate and constitutional supercooling phenomena, and may modify the smoothness of the interface. As a result, fluctuations in impurity concentration will appear in the form of growth banding.

3.15Inclusions

Tiny crystallites of the same or different species may precipitate on the growing surface of a crystal. Whether these are expelled from or occluded into the growing crystal is determined by the relation between attractive and repulsive forces with the growing crystal, similarly as partitioning of impurity elements. At higher growth rates, it is more likely that the precipitates will be occluded into the crystal. They become solid inclusions, which serve as good indicators of the chemical composition of the ambient phase and growth conditions, temperature, and pressure. Solid inclusions in natural diamond crystals have been extensively investigated, from which two types of ambient phases, ultramafic and eclogitic, have been considered as the probable ambient phases in which the diamonds were formed.

When the ambient phase is a solution phase with low viscosity, such as a hydrothermal solution, occlusion of the mother liquid (the liquid in which the crystal grows) into a growing crystal occurs at places where dendritic branches conjugate, or through overhanging of macro-growth steps, or at places where advancing macro-steps meet. When growth proceeds further on the wall of a mother–host liquid inclusion, it will dissociate into arrayed smaller inclusions by necking, or will become a negative crystal bounded by crystallographic faces.

58 Crystal growth

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