- CHAPTER 1

CRYSTAL GROWTH: THEORY AND TECHNIQUES

1. I Introduction

Crystals are the back bone of todays technological development. Present day

applications require varied types of crysTals in its purest form. This led the people

to concentrate on the development of new varieties of crystals, highly pure in form

with almost nill defect. Urgency on the development of crystals led to the

investigation of theoretical studies on its growth process. A well developed theory

is now available to help the crystal growers to perfect the growth methods. At

present theory and the experimental techniques have been developed to such a level

that tailor made crystals for specific application can be grown. This chapter gives a

general introduction to the theories and techniques of crystal growth.

1.2 The thermodynamics of crystal growth

The thermodynamic equilibrium between solid and liquid phases occur when

the free energy of the two phases are equal.

The free energy of a system is related to the internal energy and the entropy of the

system by the Gibbs [ l ] equation

G = H - T S (1.2)

where H is the enthalpy, S is the entropy and T is the temperature.

The formation of a crystal can be considered as a controlled change of phase

to the solid state. The driving force for crystallization comes from the lowering of

the free energy of the system during this phase transformation. The free energy

change associated with such a transition is

where AH = H L - H S

A S = S L - S S

AG = GL-Gs

At equilibrium AG = 0

where T, is the equilibrium temperature.

where A T = T , - T

AG is positive when T, > T and it depends on the latent heat of transition.

The free energy change can also be represented a! the product of the entropy

change and super cooling AT.

Though this representation is convenient for melt growth, one may depend on

concentrations rather than supercooling for solution growth and vapour growth.

Thus the equation modifies to

In general

where S is the supersaturation ratio. Equation (1.4) and (1.6) explain how the free

energy changes depend on the parameters like supercooling and supersahration

which are decisive in the process of crystallization. The rate of growth of a crystal

can be regarded as a monotonically increasing function of AG, if the other

parameters remain the same.

1.3 Nucleation

Nucleation process is the conglomeration of atom! or molecules to form the

first sub-microscopic speck or nucleus of the solid crystal. Nucleation can be either

homogeneous or heterogeneous. Considering the total ii-ee energy for a group of

atoms, a theory for the formation of a nucleus was put forward by Volmer and

Weber [2].

Fluctuations within the supersaturdted solution give rise to small clusters of

molecules known as "embryos". The probability that an embryo will grow to form

a stable nucleus depends on the change in free energy associated with its formation.

If the free energy change between the solid and liquid is AG,, the free energy of the

system decreases by this amount for each unit volurne of the solid created, but

increases by an amount equal to the interfacial energy o, for each unit area of the

solid-liquid interface formed. Hence the change in Gibbs free energy associated

with the fortnation of a spherical embryo of radius r is given by

A graphical representation of this equation is given in Figure 1.1 in which the

contribution due to both surface and volume to the free energy charge are

represented. The surface energy term increases with 9 and the volume energy term

decreases with ?. The net free energy change increases with the increase in size,

attains a maximum and decreases for the further increase in the size of the nucleus.

The size of the nucleus corresponding to the maximum free energy change is known

as the "critical nucleus". It is the smallest sized embryo which can grow further

effecting the reduction in the free energy of the system. On the other hand if the

size of the nucleus formed is below the critical dimension, no further growth is

possible and it will re-dissociate into the mother system.

The minimum size of a stable, critically sized nucleus is obtained by the

maximization of Equation 1.7 for r, which leads to

It may be noted that the value of r* decreases with AG, i.e., with supersaturation or

supercooling. The free energy change AG* leading to the formation of the critical

nucleus or, in other words, the activation energy necessary for nucleation, can be

calculated by substituting r* in Equation 1.7.

AG* = 3s:

Equation 1.8 can be rearranged by introducing the Gibbs-Thomson relation and

becomes

16r$@ AG* =

3(kT 1n.s)'

where Q is the molecular volume.

The rate of nucleation i.e., the number of nuclei formed per unit volume per

unit time can be expressed as

Fig. I . 1 Free energy change of ;I nucleus as a function of radius

substituting for AG*;

where J, is the pre-exponential factor. This equation shows that the nucleation rate

is governed by the temperature, degree of supersaturation and the interfacial energy.

Rearranging Equation 1.10 and arbitrarily choosing J = 1, so that 1n.J =0, we get the

expression for critical supersaturation as

Using the values of various parameters, the critical supersat~lration required for

spontaneous nucleation can be estimated.

1.4 Crystal growth theories

The successive growth of the critical nuclei of microscopic size leads to the

formation of a crystal. In order to understand the mechanism and the kinetics of

growth, several theories have been developed, which includes surface energy theory,

diffusion theory, etc., but have been found to be unsatisfactory. Later Kossel and

others analysed the atomic inhomogeneity of a crystal surface and explained the role

of step and kink sites on the growth process. However, this was also not enough to

provide a complete explanation for the continuous growth of a crystal surface. The

f d breakthrough came when Frank showed that crystal dislocations were capable

of providing the sources of steps required for the continuous growth of a crystal.

These theories have been extensively described by a number of authors 13-71.

1.4.1 S~lrface energy theory

The surface energy theories are based on the thermodynamical treatment of

equilibrium states put forward by Gibbs 111. He compared the growth of crystals

with the formation of water droplets in mist and defined the equilibrium form as one

with minimum total surface energy for a given volume. The thermodynamical

treatment suggested by Gibbs was later extended by a number of rsearchers. Curie

[81 calculated the shapes and end forms of crystals in equilibrium with solution or

vapour, consistent with Gibbs criterion. Wulff [9] gave an extension to Curie's

ideas and deduced relations connecting the growth rate of different faces and the

corresponding surface free energies. Marc and Ritzel [lo] further developed the

concepts of Wulff, stating that different faces have different soiubilities. They

suggested that when the difference in solubility is small, growth is mainly governed

by surface energy and the change in su&dce of one form is necessarily at the expense

of other. Bravais [I 11 suggested that the velocities oC the growth of the different

faces of a crystal would depend on the reticular density and the concept was

modified by Donnay and Harker [12]. Soehnke [13] proposed that the faces which

possess the greatest reticular densities are those with minimum surface energies and

hence have minimum velocities of growth.

Berthoud [14] and Valeton [IS] disputed the surface energy theory on the

basis of supersaturation. According to surface energy theory as the supersaturation

increases, the growth becomes rapid in all directions. Consequently, the crystal

habit ought to approximate to a spherical shape. But experimentally it has been

observed that when the supemturation is high, well defined faces are developed.

1.4.2 Diffusion theory

The diffusion theories, proposed by Noyes and Whimey (161 and by Nernst

( 171 is based on the following assumptions.

(a) There is a concentration gradient in the vicinity of a growing surface

(b) The growth process is a reverse process of dissolution.

According to them, the amount of solute molecules tfat will get deposited over the

surface of a growing crystal in the supersaturated solution can be written as

where dm is the mass of the solute deposited over the crystal surface of area A

during time dt, D is the diffusion coefficient of the solute, C and C, are the actual

and equilibrium concentration of the solute and 6, the thickness of the stagnant layer

adjacent to the crystal surface. But this theory also fails to be consistent with

experimental observations.

1 .4.3 Surface nucleation model

The role of surface inhomogeneities on the growth process is the basis of

surface nucleation model. A pure perfect crystal of a single element has-surfaces

covered by steps with terraces between. These steps possess kinks and hence there

are three types of sites; tenace, ledge and kink sites. This model developed by

Kossel [18], Volmer [I91 and Stranski [20] presume that crystal growth is a

discontinuous process taking place by the adsorption of matter layer by layer on the

crystal surface. According to this theory the growth units arriving on a crystal

surface do not incorporate immediately into the lattice, but become adsorbed and

migrate over the surface. The migration distance xs of the adsorbed molecule is

given by

where Ds is the surface diffusion coefficient and T, is the mean life time of an

adsorbed molecule on the surface. In this model

x, - a exp - I 2kT I where a is the nearest neighbour distance and $J is the nearest neighbour interaction

parameter.

The possible lattice sites for the attachment of adsorbed atoms on crystal

surface is illustrated in Figure 1.2 in which adatoms are pictured as cubes.

Fig. 1.2 Possible lcttice sites for the i~ttac,hment of absorbed atom

A - terrace site

B - ledge site

C - kink site

An atom reaching a kink site is attracted by three out of the six nearest

neighbows, while such bonds is two for a ledge and one for a terrace site. Thus the

maximum binding energy between the adatom and the existing crystal surface occurs

for the incorporation at a kink site. Hence the adatoms over the crystal surface

migrate towards a step and moves along it to a kink site and get incorporated.

Under ideal conditions this stepwise stacking will continue until the whole layer is

completed. The mode of advance of a step has been extensively analysed by Burton,

Cabrera and Frank 1211. A step advances by incorporating more and more adatoms

at kmk sites. Assuming a diffusional flow of adatoms along the step, the rate of

advance of a step can be expressed as

where y i s a frequency factor a d W is the total evaporation energy. The rate of

advance is then proportional to supersaturation S and to the mean migration distance

x,. Here it is assumed that the mean migration distance is much higher than the

mean distance between two adjacent kink sites on a step.

When an advancing step covers the whole surface completely, further growth

is possible only by the initiation of a two dimensional nucleus. According to

Volmer this is possible on account of thermal fluctuations. Assuming a circular

disc-shaped nucleus of radius r and height h, the free 'znergy change associated with

the formation of such a two dimensional nucleus may lie written as

where ?- is the edge free energy. Employing the same techniques adopted in

Section 1.3 the activation energy necessary for the hvo dimensional nucleation can

be calculated as

*hP0 AG*, = -

kT ins

and the nucleation rate is given by

The expression for critical supersaturation is given by

The rate of growth of a singular face is in principle controlled by the rate of

nucleation and rate of advance of a step and can be expressed as

The growth rates calculated using these equations do not agree with observed results

in many cases. According to the K.S.V. theory, the growth of the crystal is

controlled by the probability of two dimensional nucleation which is not appreciable

until the supersaturation reaches a considerable percentage order. However, it is

observed that most of the real crystals grow at supersaturation down to a value of

1 % or even lower.

1.4.4 Screw dislocation theory

The discrepancy between the observed growth rate and the theoretical

prediction based on two dimensional nucleation theory points to the fact that there is

some other mechanisms responsible for the continuous growth of a crystal surface.

Frank 1221 proposed that dislocations having a screw component can act as a

continuous source of steps on the surface of the crystal which eliminates the need for

surface nucleation. A screw dislocation emerging at a point on the crystal surface

provides a step on the surface with a height equal to 'a', the projection of the

Burgers vector of the dislocation. Since the step provided by the screw dislocation

is anchored at the emergence point of the dislocation, and since the inner parts of the

steps move radially at a faster rate than the outer parts, fiuther growth takes place

only by the rotation of step around the dislocation point. This mechanism is

illus~rated in Figure 1.3. Under a given condition of supersaturation these steps wind

themselves up into a spiral, centred on the dislocation. Though this theory assumes

the existence of dislocations in the crystal in order to enhance the growth of a

surface, the growth rate does not depend on the concentration of these dislocations.

Fig. 1.3 Developm~:nt of a spiral

Based on spiral growth mechanism Burton, Cabrera and Frank could establish a ,

relation between the rate of growth R and the relative supersaturation which is

expressed as

R = C (s2/s,) tan h (slls) (1.21)

whe:re the parameter sl is defined as

and D, fl ,BQ

C =

x,2

where s - relative supersaturation

S1 - a constant for BCF model

rise - equilibrium concentration of growth units on surface.

B - retardation factor

Q - volume of the growth unit.

The variation of the growth rate with supersaturation thus depends on two

parameters, C which determines the absolute value of growth rate and sl which

determines actual growth rate.

The BCF theory therefore predicts that the growth rate 15 proportional to the square

of the supersaturation for low supersaturation, changing to a linear dependence at

higher supersaturations. The Frank model is currently well founded in respect of the

excellent agreement between theory and observations on growth rate as well as of

the direct observation of spiral pattan, characteristic of this mechanism 123-271.

However, Keller [28], and Bauser [29] observed that edge dislocations can also act

as persistent sources of steps for crystal growth. Frank [30] proposed a general

explanation of nucleation at edge dislocation that the surface stresses provides the

extra energy required for the formation of the growth nuclei when the dislocation

component perpendicular to the surface is absent.

1.5 Crystal growth techniques

The process of crystal growth is a controlled change of state, or phase

change, to the solid state. This transition may occur from the solid, liquid or vapour

state. Depending on the phase transitions involved in the process, the crystal growth

methods can generally be classified into four main categories 131-341.

I ) Solid growth (solid -r solid)

2 ) Vapour growth (vapour -, solid)

3) Melt growth (liquid -r solid)

4) Solution growth (liquid -, solid)

The growth of crystals from the liquid phase is treatetl as two categories due to the

independent behaviour of melt and solution techniqu't:~. A brief description about

the various techniques of growth has been presented in the following sections with

an emphasis to liquid - > solid growth method.

1.5.1 Solid growth techniques

In solid growth technique, single crystals are developed by the preferential

growth of a polycrystalline mass. This can be achieved by straining the material and

subsequent anneal'ig. Large crystals of several materials, especially metals have

been grown by this method [35]. The main advantage of solid growth method is

that this technique permits the growth at low temperatures without the presence of

additional component. But as the growth takes piace in the solid, density of sites for

nucleation is high and it is difficult to control nucleation.

1 -5.2 Growth from vapwr

Vapour growth techniques can be adopted for the growth of materials which

lack a suitable solvent and sublime before melting at normal pressure. Vapour

growth methods have been employed to produce bulk crystals and to prepare thin

layers on crystals with a high degree of purity. Growth from vapour phase may

generally be subdivided in to

1) Physical vapour hamport

2) Chemical vapour tramport.

Physical w q o w rrmpon (PW)

In PVT technique the crystal is grown from its own vapours and this method

does not involve any extraneous compound formation or reaction. The PVT

methods are limited to materials having an appreciable vapour pressure at attainable

temperatures. There are two types of techniques employed in physical vapour

transport process-sublimation-condensation and sputtering. The first method

involves sublimation of the charge at the high temperature end of the furnace,

followed by the condensation at the colder end [36,37]. Sputtering techniques are

preferred to low vapour-pressure substances and mainly this method has been used

to prepare thin films rather than discrete crystals. The principal advantage of this

technique is that film growth can be possible at lower temperature than in ordinary

sublimation-condensation growth. The PVT techniques are used to prepare a variety

of crystals [ 3 8 4 ] and for the production of epitaxial films [41,42].

Chemical vapour trmpn

Chemical vapour transport technique involves a chemical reaction between

the source material to be crystallized and a transporting agent. The material to be

crystallized is converted into one or more gaseous prtduct, which either diffuses to

the colder end or gets transported by a transporting (carrier) gas. At the cold end,

the reaction is reversed so that the gaseous product decomposes to deposit the parent

material, liberating the transporting agent which diffuses to the hotter end and again

reacts with the charge. The commercial importance of vapour growth is in the

production of thin layer by chemical vapour deposition [43-471.

1.5.3 Melt growth technique

Melt growth is the process of crystallization by fusion and resolidification of

the pure material. It is the fastest of all crystal growth methods and is widely used

for the preparation of large single crystals. Melt g~owth methods are limited to

materials which melt congruently and having an experimentally viable vapour

pressure at its melting point. This method requuc:~ only simple systems. The

material to be grown is melted and after that it may progressively cooled to yield the

crystalline matter. This method has been generally employed for the growth of

metals, semiconductors, and laser host crystals. Single crystals with high degree of

perfection and purity can be obtained by this method.

Usually melt growth methods are grouped into two categories.

1) N o d freezing method

a) Bridgman technique

b) Cmhralski technique

2) Zone-growth method

a) Zone melting method

b) Floating zone method

There are two versions for Bridgman's method; Horizontal Bridgman method

(Chalmer's technique) and Ver&ical Bridgman method (Bridgman-Stockbarger

technique). In these techniques directional solidification is obtained by slowly

withdrawing a boat containing molten material through a temperature gradient

148,491. The Bridgman technique is most frequently applied for the growth of

metals, semiconductors and alkaline earth halides [5@!;2]. But this method cannot

be used for materials having high melting point and high expansion coefficient.

Cmhralski method is the most powerful method for growing single crystals

and is basically a crystal pulling system. The advantage of this method over the

Bridgman method is that it can accommodate the volume expansion associated with

the solidification. Cmhralski method has gained wide recognition particularly in

growing single crystals of semiconductors like silicon [53] and other materials

154,551.

Zone melting is mainly considered as a purification technique. However, ~t

may be used as a method for the growth of single c rys~ls . In this method, a zone

or part of the solid material is melted and this molten zone travels together with the

heating elements. The advantage of zone melting is that it offers a relatively simple

way of producing doped crystals containing deliberately admixed additives in a

given concentration in uniform distribution 1561. Floating zone technique developed

by Keck and Golay 1571 is a variant of the zone melting technique in which no

crucible is used. This method is especially suitable for the preparation of high

purity silicon and germanium.

1.5.4 Growth from solution

The growth of crystals by precipitation from aqueous solution is the most

simple and oldest technique. In this process, a saturated solution of the material in

an appropriate solvent is used from which crystallization takes place as the solution

becomes critically supersaturated. The supersaturation can be achieved either by

lowering the temperature of the solution or by slow evaporation. The advantage of

the method is that crystals can be prepared from a solution at temperatures well

below its melting point, perhaps even at room temperature and therefore it turns out

to be more applicable in many cases [58].

Solution Growth can be broadly classified into

1) High temperature solution growth.

2) Hydrothermal method.

3) Low temperature solution growth.

4) Gel growth

High temperature solution growrh

Flux growth is the term used to describe the growth of crystals from molten

salt solvents at high temperatures. A high temperature solvent which reduces the

melting temperature of the solvent is referred as flux [59]. This reduction in

temperature is probably, the main advantage of flux growth. The materials to he

crystallized are dissolved in a proper solvent at a temperature slightly above the

saturation temperature and then slowly cool the crucible so that growth occurs in a

spontaneously formed nucleus. The growth of crystals by the slow cooling of the

flux is also effective.

Hydrothermal methods

Hydrothermal method is suited for the growth of certain class of materials

which are practically insoluble in water. A number of metals and metal oxides show

an appreciable increase in solubility when the temperature and pressure are

increased.

It can be treated as aqueous solution growth at elevated temperature and

pressure. Growth is usually carried out in steel autoclaves with gold or silver

linings. A charge of crystals is dissolved in the lower part of the autoclave. The

hot saturated solution is directed towards the upper (colder) part, where it cool and

become supersaturated hence the growth of crystal. The spent solution returns to

the other part and this process continues until the whole charge is recrystallized.

The solution simply acts as a transporting agent for the solid phase. This method is

extensively used for the growth of large high quality synthetic quartz crystals 1601.

Low rempermre solurion growth

It is the most effective and easy way for growing crystals. A variety of

crystals can be grown by this technique at room temperature. In this method, a

saturated solution of the material is prepared in a suitable solvent and crystallization

is initiated by slow cooling of the solution or by slow evaporation of the solvent.

Large and perfect crystals of industrially important materials are grown by this

method [6 1-65].

The gel method

All the methods mentioned above fail in the case of the growth of certain

class of materials having poor solubility and unstable thermal behaviour. The gel

method is found suitable for the growth of such a class of materials. In gel

technique supersaturation is achieved either by the slow interdiffusion of solutions of

two reacting species, which on mixing react to form the solute, or by the

interdiffusion of a solution with a solvent in which the solute is insoluble or less

soluble. Using gel technique small but highly perfect crystals of relatively insoluble

materials can be grown [66]. This technique fonns the subject matter of this thesis

and therefore needs further elaboration. It is given in the following chapter.

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