25 Crystal growth theory

Prof. P. N. Kotru

TABLE OF CONTENTS

25.1 Equilibrium Form of Crystals

25.2 Wulff’s Law

25.3 Ideal Crystal Growth

25.4 Real Crystal Growth

LEARNING OBJECTIVES

• In this module we shall learn about some theoretical aspects of crystal growth.
• Equilibrium form of a crystal on the basis of Kossel model is discussed.
• Simple cubic structure is considered in the discussion on determination of equilibrium form of crystals .
• Wulff’s theorem is explained.
• The cube model to discuss ideal crystal growth is considered and understanding of how a two-dimensional nucleus grows into a complete layer is given.
• It is explained that actually real crystal growth does happen under conditions in which two-dimensional nucleation is impossible.
• The growth of real crystals is explained on the basis of Frank’s model of screw dislocation theory which provides a source of never ending repeatable steps, thus accounting for the growth of real crystals at low
• Crystal growth due to a step provided b y the intersection of screw dislocation on the surface of the crystal which give rise to a crystal growth in a spiral form is explained using schematic illustrations.

25.1     Equilibrium Form of Crystals

Kossel in 1927 used a very simp le model of a crystal to calculate relat ive surface free energies. The objective is to determine equilibriu m form of a crystal and in order to achieve it one is required to find binding energy between the atoms. It is well known that the atoms of a crystal are arranged in a definite geo metrical pattern in which each unit cell is periodically repeated and each atom has specific structural surroundings. The nucleation and growth processes of crystals depend strongly on the crystal structure. However, in the discussion on equilibriu m form of crystals we will use the simp le cubic structure as shown in figu re 25.1 in which each represents one ato m and also one un it cell.

Figure 25.1: Different atomic sites on the surface of a simp le cubic structure

In this figure is shown an incomplete face and there are various atomic sites designated by numbers with differing bonds of strength. Some atomic sites of significance may be identified here in the figure itself. Such sites are:

Site marked 1 is face vacancy,

Site marked 2 is edge vacancy,

Site marked 3 is adsorbed atom,

Site marked 4 is co mpleted step,

Site marked 5 is step edge,

Site marked 6 is incomp leted corner,

Site marked 7 is incomp leted edge,

Site marked 8 is co mpleted surface,

Site marked 9 is co mpleted corner,

Site marked 10 is co mp leted edge,

Site marked ½ are half-crystal posit ions, also kno wn as kink sites.

In the simplest form of crystals the binding energy between the atoms varies with sixth power of the distance between them. To understand this one may refer to figure 25.2, wh ich shows a simple cube model in which atoms are represented by cubes. The atoms (cubes) are labelled as 1, 2, 3 and 4 and rep resenting first, second and third-nearest neighbours of an ato m (cube).

Figure 25.2: Cube model showing first, second and third nearest neighbours of an  atom

Suppose he centres of the two cubes which share one face are separated by a distance d. In other words, the distance between an atom and one of its nearest neighbours say, as for example,

1 and 2 is d. The distance between the second nearest neighbours (cubes) joined at one edge, say 1 to 2, is √2.d. The distance between the third nearest neighbours, say 1 to 4 of figure 25.2 is √3 .d. No w, if t wo cubes (atoms) are b rought fro m infinity to a distance d fro m each other, the potential energy released is ф/d6 , for the second nearest neighbouring cubes (atoms ) ф /8d6 and for the third nearest neighbouring cubes ф /27d6 and so on . Since the number of neighbouring atoms at a distance say , ‘r’ increases as r2, while the force decreases as r6 , those neighbouring atoms which are distant farther away do not influence the total b inding energy much and as such should be ignored .

Now let us go to the interior of the crystal. Consider a cube in the interior of the crystal, it has six (6) nearest, twelve (12) second nearest and eight ( 8 ) third nearest neighbour. As argued above, neighbouring cubes which are farther away are ignored for reasons given above.

Therefore, the b inding energy is 6 ф /d6  + 12 ф /8d6 + ф /27d6

The binding energies of cubes will depend on the position it is occupying on the surface. Referring to figure 25.1 the cubes are shown at different positions and are accordingly labelled by nu mbers. So me have one of their sides attached to only one face of the cube under it (e.g., number 3), five faces of the cube attached to the faces of cubes on all the sides except the top face (e.g., number 8) and so on. So, the binding energies of each cube will depend on where it is placed. The interesting position is the” kin k site”, labelled as ½ , and also known as ‘ half-crystal’ site as shown in figure 25.1. The bind ing energy of a cube at the kink site is exactly half the total binding energy of an interior atom. The addition of an atom to a half-crystal position creates another half-crystal position to which an atom can be added. On a kinked step, atoms can continually be added to the half-crystal positions on the plane and create new half-crystal positions as the crystal grows plane by plane. On a kinked step, atoms can get attached at the kink site until the row of ato ms gets completed. Except for the first atom of each row, a complete plane can be built up by successive additions at half-crystal positions. The relative b inding energy of a cube at the kin k site is given by:

3 ф /d6 + 6 ф /8d6  + 4 ф / 27d6

This is exactly one – half of the binding energy of an interior cube.

At equilibrium, an atom at the’ kink site’ (1/ 2 site) has the same probability to evaporate as is for an ato m to condense at the adjacent empty half-site. It is easier to remove cubes (atoms) which are not that strongly bonded as those positioned at the half-site and ,therefore, cubes (atoms) get added up in sites where they would get as strongly bonded as the one at the kink site. This This is how the building of equilibrium form of a crystal may be thought of. However, this method does not directly lead us to the equilibrium shape of a crystal. It also does not provide any lead to knowledge regarding ratio of the areas of various faces at equilibrium. Using certain appropriate assumptions, it is used to calculate interfacial energies for different faces which can then be used alongwithWulff’s law (described below) to provide a lead in giving equilibriu m shape of a crystal.

25.2 Wulff’s  La w

Wulff’s theorem states that in a crystal at equilibriu m the distances of the faces from the centre of the crystal are proportional to their surface free energies per unit area.

Wulff’s law is aimed at determin ing equilibrium form of a crystal from known interfacial energy through geometrical construction. The interfacial free energy of every possible face is taken as a vector originating fro m the origin with length of each vector made proportional to the interfacial free energy and directed perpendicular to the face, is plotted. The curve so drawn connecting the ends of these vectors lead to formation of cusps. Each face is then drawn at the position of its cusp and the equilibriu m form is given by the interior figure so obtained. This is known as Wulff’s construction of equilibriu m form of crystals. This construction is precisely correct and as such assumes great theoretical interest and importance. The crystal model as given by Kossel offers a realistic picture of crystal surfaces only at low temperatures.However, at high temperatures , the thermal v ibrat ional energy of the cubes (atoms) is called into play which cannot be ignored and has to be included.

25.3   Ideal  Crys tal Growth

To explain crystal growth we will consider the cube model shown in figure 25.3.

Figure 25.3: Schematic  diagram  showing cube model

We start with a perfect crystal which is in its equilibriu m form. Under isothermal conditions growth of the crystal will take place if it is subjected to a supersaturated environment .The process is accompanied by a decrease in free-energy, and the crystal struggles for growth. The growth has to happen atom by atom or molecule by molecule. In the cube model of figure 25.3, each atom or molecule is represented by a cube. For the addition of first atom the most preferable site is the centre  of one of the faces.However, the attachment at this site is not very stable as it is bonded less strongly as compared to the kink site. At the kink site or the half-crystal site the atom or molecule is much more strongly bonded than any other site. Unless there are thermal fluctuations the process of crystal growth cannot start. Figure 25.3, a cube model, shows the structure of a face under a situation where thermal fluctuations exist. Single cube (atom) at an isolated site on the surface is far less stable than two cubes (atoms) in contact. The cube (atom) at the kink site is far more stable than anywhere else. The process of transitory groupings of additional atoms and of vacancies (holes) exists. In this way, the crystal face after complet ing a layer, must add complete new layers. The initiation of a new fresh layer is a problem of nucleation. The problem is that of what is called “two-dimensional nucleat ion” The growth of nucleus in this case takes place by a process shown in figure 25.3.

The process  is  that of migrat ion  of cubes  to sites  where they  are more strongly  bonded.  A  process which involves migration of ato ms say at the place labelled number 1 to position 1/ , 2 to 2/ , 3 to 3/ and so on. This is how a layer keeps on building up.

For a t wo-dimensional nucleus in the form of d isc of radius r and height d, the free energy of format ion is:

∆F = ( fC ─ fE) π ( ∆r2 ) d + 2 σ π ( ∆r) d ,

Where ( fC─ fE ) represents free energy change in crystallization and σ is the surface free energy. The subscript E here in the above expression stands for “Environment”. In the formation of this disc, the excess surface grown is circu mference times the height d and the critical radius r* associated with the

maximu m value of ∆F (i.e., F*) is  σ.( f_E─ f_C )

In considering growth of crystals, it is important to understand how a t wo-dimensional nucleus grows into a complete layer. Any incomplete step has a kink associated with it and atoms migrating to kink sites are stable. This way, the step becomes complete as fast as surface-atoms migrate to kink sites and get attached there. Thermal fluctuations create situations that lead to a steady concentration of kinks in any step. Nucleation of new rows of atoms can get so rapid that it is of no consequence in step growth.

A perfect crystal cannot grow at all until the supersaturation is more than some crit ical value. Vo lmer and Schultze performed experiments to test two-dimensional nucleation growth theory by examining the growth of iodine, phosphorous and napthalene as a function of supersaturation. They observed that the growth rate was much higher for iodine at saturation value far below what was expected from the two-dimensional nucleation theory. They found that the growth rate of iodine was high at saturation values at which they expected it to be negligible. Volmer and Schultze’s experimental results required an alternative source of growth centres.The explanation offered by Frank was screw dislocation growth discussed in the section that follows.

25.4   Real Crys tal Gro wth

Real crystals grow under conditions in which two-dimensional nucleation is impossible. The creat ion of a step on perfect crystal surface due to coalescence of several isolated atoms which are rando mly deposited from the liquid or vapour phase is an event of very low probability, because such atoms are not very strongly bound to the crystal face and do not usually remain attached to the crystal long enough to join up to formation of a growth step. These atoms are usually removed by thermal agitation before they are able to do so. However, if a step is formed in this manner it will then rapid ly spread across the crystal face by the addition of atoms at the edges, where they are strongly bound. This may continue until a new atomic layer has been added. Further growth then takes place only with great difficulty, since a new step is required to be formed on the added layer.

It was Frank who suggested that screw dislocations having a displacement vector parallel to the crystal face would provide a source of steps, which wou ld persist and never disappear. Screw d islocation becomes a source of repeatable steps.

In order to understand as to how screw dislocation becomes a source of perpetual step with the help of which the crystal can grow, we need to relook at the format ion of this type of dislocation. Consider a crystal in wh ich a cut XYZK is made partway on its surface and one of the two sides of the cut is pressed so as to cause their displacement relative to each other by one atomic spacing in the direction shown by the arrows in figure 25.4 (a).Since the displacement is by exact ly one atomic spacing , the vertical faces of the crystal stay perfect and the imperfection is confined to the cylindrical volu me X-K as shown in figure 25.4 (b ).The steps X-Y and Z-K can never disappear either as a result of growth or by dissolution. The top and bottom surfaces will always remain imperfect and can never be perfected. It is much easier for a crystal to grow by diffusion of ext ra ato m to the step of a screw dislocation than by spontaneous addition of atoms to perfectly flat crystal surfaces.

Addition of atoms at the step of a screw dislocation cannot destroy it, it simp ly sweeps round the actual point at which the screw dislocation intersects the crystal face rather like the beam o f a light house which sweeps round.

Figure 25.5 is a schematic diagram showing crystal growth at a step anchored at a screw dislocation which results in a spiral fo rmation. On account of the fact that the step X─Y is anchored at point X and because the step will grow by surface diffusion at the same rate during growth this step must wind itself up around the point X into a spiral as shown in figure 25.5. This is how Fran k’s model of real crystal growth resolves the fundamental discrepancy between theory and experiment . Most real crystal growth does not require two-d imensional nucleation and spreading of layers.

This is how spiral growth was predicted on theoretical reasoning by Frank in 1949. Practically, spiral pattern of the growth was first observed by Griffin in 1950 on the surface of a beryl crystal. Later spiral patterns were also observed on large number of crystals like SiC crystal (Verma), quartz among many others. Through measurements it has been shown that the height of the steps grown is of one atomic or mo lecular size .

Figure 25.5 (a,b,c,d) : Schematic illustrat ion of crystal growing in a spiral form at the edge of t he discontinuity X Y as shown sequentially by stages indicated in a,b,c & d as growth progresses

Summary

• Theoretical background of how crystals grow is considered.
• Equilibriu m form of a crystal on the basis of Kossel model is discussed.
• Equilibriu m form of a crystal is discussed on the basis of simp le cubic structure .
• Wulff’s theorem providing important inputs in determining the equilibriu m form of crystals is explained.
• Process involved in nucleation and completion of layer on the surface of a perfect crystal is discussed.
• It is explained that actually real crystals growth occurs under conditions in which two dimensional nucleation is possible.
• Growth of crystals at low temperature  is explained on the basis of Franck’s model of screw d islocation theory.
• It is exp lained that screw dislocation emerging on a crystal surface provides a perpetual step which never ends, as a result of which adsorption of atoms along the step occurs at lower temperatures and growth of crystal can take place continuously.
• This  type of growth leading to formation of spiral steps on the growing crystal surfaces  is exp lained and illustrated with the help of a schemat ic diagram.
• The screw dislocation theory of a crystal growth is supported by observation of spirals surfaces by several researchers.