theories and methods of crystal growth: a general review · crystal growth: a general review 1.1...

Chapter 1

Theories and Methods of Crystal Growth: A General Review

1.1 Introduction

The growth process of single crystals has been developed over the

years to meet the requirements of basic research and technological

applications. Crystals play a significant role in electronics industry,

photonics industry and fibre-optic communications. The semiconductor

based electronics industry, requires high quality semi-conducting,

ferroelectric, piezoelectric, single crystals. Integrated micro electronics

and opto-electronics have necessitated improvements in crystal growth

techniques. The developments in the theory and techniques of crystal

growth have made possible the commercial scale growth of large, defect-

free crystals of silicon and GaAs which find application in the fabrication of

electronic, photonic and microwave devices.

The crystal growth process is a complex one as it involves

optimization of various parameters such as temperature, pressure,

chemical potential, electrochemical potential etc. In a well-defined growth

process, just one of these parameters is held minimally away from its

equilibrium value to provide a driving force for growth. The process of

crystal growth involves phase transitions of the following types: solid-solid,

liquid-solid and gas-solid. The methods of crystal growth are classified

accordingly. A general review of the theories and techniques of crystal

growth is given in this chapter.

2 Chapter 1

1.2 Thermodynamics of crystal growth

Crystal growth being a non-equilibrium process, control of the

crystal growth environment and a consideration of growth kinetics, both at

the macroscopic and atomic levels are significant. The phase

transformation during the crystal growth process occurs due to lowering of

the free energy of the system. The free energy is related to the entropy

and internal energy of the system by the Gibbs [1] equation

G = H – TS (1.1)

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

The formation of a crystal can be considered as a controlled

change of phase to the solid state. The driving force of crystallization

comes from the lowering of the free energy of the system during this

phase transformation. This change of free energy is

∆G = ∆H – T. ∆S (1.2)

where ∆H = HL - HS

∆S = SL - SS

∆G = GL - GS

At equilibrium, ∆G = 0

∴ ∆H = Te. ∆S

where Te is the equilibrium temperature.

∴ ∆G = ∆H . ∆T/Te (1.3)

where ∆T = Te – T

When Te > T, ∆G is positive and it depends on the latent heat of transition.

The free energy change can also be represented as the product of

the entropy change and supercooling ∆T.

∆G = ∆S. ∆T

Theories and methods… 3

This representation is convenient for melt growth. In the case of

solution growth and vapour growth, what is taken into account is

concentration rather than supercooling. Hence this relation becomes

∆G ∼ R T ln (C/Co)

∆G ∼ R T ln (P/Po)

where C, P are solute concentration and vapour pressure and C0, P0 are

equilibrium solute concentration and saturation vapour pressure.

In general

∆G ∼ R T ln S (1.5)

where S is the supersaturation ratio.

The rate of growth of a crystal can be regarded as an increasing

function of ∆G, if the other parameters remain the same.

1.3 Nucleation

Nucleation is the precursor of crystal growth and the overall

crystallization process. In a supersaturated or supercooled system, when

a few atoms or molecules join together, a change in free energy takes

place in the process of formation of a cluster in the new phase. The

cluster consisting of such atoms or molecules is called a nucleus. If the

nucleus grows to a particular size known as ‘critical size’, then there is a

greater probability for the nucleus to grow. There are four stages involved

in the formation of a stable nucleus:

(1) the development of saturated stage, which may be due to a

chemical reaction, a change in pressure, temperature or any other

chemical or physical condition.

(2) the generation of nucleus.

(1.4)

4 Chapter 1

(3) the growth of nucleus from the unstable state to stable state or to

the critical size.

(4) the relaxation process where the texture of the new phase

changes.

Nucleation may occur spontaneously or it may be induced

artificially. Based on these, nucleation is classified as follows:

(1) Homogeneous nucleation and

(2) Heterogeneous nucleation.

The spontaneous formation of crystalline nuclei in the interior of the

parent phase is called homogeneous nucleation [2].

If the nuclei form heterogeneously around ions, impurity molecules

or dust particles, it is called heterogeneous nucleation [3].

Considering the total free energy for a group of atoms, a theory for

the formation of a nucleus was put forward by Volmer and Weber [4].

The free energy change associated with the formation of a nucleus

can be written as

∆G = ∆Gs + ∆Gv (1.6)

where ∆Gs is the surface excess free energy of the interface separating

the parent and the product phases, and ∆Gv is the volume excess free

energy change per unit volume, which is a negative quantity.

For a spherical nucleus

∆G = 4πr2σ –

4

3 πr3 ∆Gv (1.7)

Free energy of the system decreases by ∆Gv for each unit volume

of the solid created but increases by an amount equal to the interfacial

energy σ, for each unit area of the solid-liquid interface formed.


A graphical representation of equation (1.7) is given in Figure 1.1.

The surface energy term increases with r2 and the volume energy term

decreases with r3. The net free energy change increases with increase in

size, attains a maximum and decreases for 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 size of

the nucleus which can grow further. If the size of the nucleus is below the

critical dimensions, no further growth is possible and it redissociates into

the mother system.

Figure 1.1 Curve showing the dependence of the change in surface energy

(∆Gs), volume energy (∆Gv) and the net free energy (∆G) on radius (r) of the nucleus

By setting the condition d

dr∆G = 0, the radius r* of the critical

nucleus is obtained as

r* = v

2σ

∆G (1.8)

∆∆∆∆G*

r

∆∆∆∆G

r*

∆∆∆∆Gs

∆∆∆∆Gv

0

Fre

e en

ergy

ch

ange

∆∆ ∆∆G

6 Chapter 1

Substituting the value of r* in equation (1.7), the free energy change

associated with the critical nucleus is given by

∆G* = 3

2

v

16 σ

3∆G

π (1.9)

According to Gibbs - Thomson relation

ln 0

C

C

= ln S = 2 σ Ω

k T r (1.10)

where k is the Boltzmann’s constant and Ω is the molecular volume.

Using this relation equation (1.9) can be written as

∆G* = ( )

3 2

2

16πσ Ω

3 k TlnS (1.11)

The rate of nucleation J, ie., the number of nuclei formed per unit

volume per unit time can be expressed as

J=J0 exp -∆G*

kT

Substituting for *G in this equation,

J = J0 exp 3 2

3 3 2

-16πσ Ω

3k T (lnS)

(1.12)

where J0 is the pre-exponential factor.

This equation shows that the rate of nucleation is controlled by

temperature T, degree of supersaturation S and interfacial energy σ.

Rearranging equation (1.12) and arbitrarily choosing J = 1 so that

ln J = 0, the expression for critical supersaturation becomes

Scri = exp

1/ 23 2

3 3

0

16πσ Ω

3k T lnJ (1.13)


Using the values of various parameters, the critical supersaturation

required for spontaneous nucleation can be estimated.

For heterogeneous nucleation, the presence of a foreign substrate

induces nucleation at supersaturations lower than that required for

spontaneous nucleation. The free energy change associated with the

formation of critical nucleus under heterogeneous conditions ∆G*het must

be less than the corresponding free energy change ∆G*hom associated

with homogeneous nucleation.

∆G*het = φ ∆G*hom

where the factor φ is less than unity.

1.4 Theories of crystal growth

The process of crystal growth involves the following steps:

generation of reactants, transport of reactants to the growth surface,

adsorption at the growth surface, nucleation, growth, and removal of

unwanted reaction products from the growth surface. When a crystal

nucleus attains the critical size, it grows into crystal of macroscopic

dimension with well developed faces. To understand the kinetics and

mechanism involved in the process of crystal growth, many theories such

as surface energy theory, diffusion theory and surface adsorption theory

have been proposed.

Gibbs proposed a theory by considering the growth of a crystal

analogous to the growth of water droplet in mist. Later Kossel [5] 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 theory could not provide a complete explanation for the continuous

growth of a crystal surface. A complete explanation for continuous growth

at low supersaturation was given by Frank [6]. He showed that crystal

dislocations were capable of providing the sources of steps required for

8 Chapter 1

the continuous growth of a crystal. These theories have been extensively

described by many authors [7-9].

1.4.1 Surface energy theory

According to the surface energy theory proposed by Gibbs, a

growing crystal assumes the shape which has minimum energy. The

thermodynamical treatment suggested by Gibbs was later extended by a

number of researchers. Curie [10] calculated the shapes and end forms

of crystals in equilibrium with solution or vapour, consistent with Gibbs

criterion. He suggested that when the volume free energy per unit volume

is constant, the sum of the surface energies of all faces of the crystal will

be minimum.

Wulff [11] deduced the relation connecting the growth velocity

measured normal to any surface and the surface energy of that surface

and found that these two are proportional to each other. Marc and Ritzel

[12] further developed the concepts of Wulff, stating that different faces

have different solubilities. They suggested that when the difference in

solubility is small, growth is mainly governed by surface energy and the

change in surface of one form is necessarily at the expense of the other.

Berthoud [13] and Valeton [14] disputed the surface energy theory on the

basis of supersaturation. According to the theory, as the supersaturation

increases, growth becomes rapid in all directions and this results in the

spherical shape of the crystal. But, experimentally, it has been observed

that well defined faces are developed when the supersaturation is high.

1.4.2 Diffusion theory

The diffusion theories, proposed by Noyes and Whitney [15] and by

Nernst [16] are based on the following assumptions.

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

surface.

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


The amount of solute molecules that will get deposited over the

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

0

dm D = A (C - C )

dt δ (1.14)

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 C0

are the actual and equilibrium concentration of the solute and δ is the

thickness of the stagnant layer adjacent to the crystal surface. The value

of δ depends on the relative motion between the crystal surface and the

solution. But this theory also fails due to the inconsistency with the

experimental results.

1.4.3 Adsorption layer theory by Kossel, Stranski and Volmer (KSV theory)

The role of surface discontinuities as nucleation sites was first

recognized by Kossel [17], Stranski [18] and Volmer [19]. According to

this theory the growth units approaching a crystal surface do not

incorporate immediately to the lattice, but become adsorbed and migrate

over the surface. The possible lattice sites for the attachment of adsorbed

atoms on the crystal surface are terrace, ledge and kink site as shown in

Figure 1.2.

Figure 1.2. Possible lattice sites for the attachment of adsorbed atom A-terrace site, B- ledge site, C-kink site

10 Chapter 1

The binding energy between an adatom and existing lattice

increases from terrace to ledge to kink site.

Volmer first suggested the concept of crystal growth mechanism

based on the existence of adsorbed layer of solute atoms or molecules on

a crystal face. Volmer’s theory (Gibbs-Volmer theory) is based on

thermodynamic reasoning. When units of the crystallizing substance

arrive at the crystal face, they are not immediately integrated into the

lattice, but merely lose one degree of freedom and are free to migrate

over the crystal face (surface diffusion). Hence there will be a loosely

adsorbed layer of integrating units at the interface, and dynamic

equilibrium is established between this layer and the bulk solution. The

adsorption layer plays an important role in crystal growth, secondary

nucleation and precipitation phenomena.

At the ‘active centres’ where the attractive forces are the greatest,

atoms, ions or molecules are linked into the lattice and under ideal

condition this stepwise build-up will continue until the whole plane face is

complete. Before the crystal face can continue to grow, a centre of

crystallization must come into existence on the plane surface and Gibbs -

Volmer theory suggests that a monolayer island nucleus, usually called a

two-dimensional nucleus, is created.

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. The free energy change associated with the formation of

such a two-dimensional nucleus may be written as

∆G = a γ − V ∆Gv (1.15)

where a and V are the area and volume of the nucleus. Assuming a

circular disc-shaped nucleus of radius ‘r’ and height ‘h’ the equation

becomes

∆G = 2 π r h γ − π r2 h ∆Gv (1.16)


where γ is the edge free energy. The activation energy for two-

dimensional nucleation can be calculated as

∆G* = 2

π h γ Ω

k T Sln (1.17)

The rate of two - dimensional nucleation can be expressed as

J′ = C exp

-∆G*

kT

= C exp 2

2 2

-πh γ Ω

k T lnS

(1.18)

The expression for critical supersaturation is given by

Scri = exp 2

2 2

πh γ Ω

k T lnC

(1.19)

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

R = 1/3 2/3h J V′ (1.20)

According to KSV theory, the growth of a 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 by Burton, Cabrera and Frank (BCF theory)

One major drawback of KSV theory is that once the kinked ledge

has received sufficient ad atoms to move it to the edge of the crystal, it

could no longer function as a low energy nucleation site. But if points of

dislocations with screw components (screw dislocations) at crystal surface

are present they can provide a continuous source of steps which can

12 Chapter 1

propagate across the surface of the crystal. A theory of crystal growth

including the mechanism of step generation and transport into the steps

was given by Burton, Cabrera and Frank [20].

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 step move radially at a faster rate than the

outer parts, further growth takes place only by the rotation of step around

the dislocation point. This mechanism is illustrated in Figure 1.3. Under a

given condition of supersaturation these steps wind themselves up into a

spiral, centred on the dislocation.

Figure 1.3. Development of Spiral

A relation between the rate of growth R and the relative super

saturation was given by Burton, Cabrera and Frank. It is expressed as

R = C 2

1

S

S tanh 1

S

S (1.21)

where S1 = s

γ

k T X

19

2

Ω


and C = s se

2

s

D

X

β Ω∩

where γ - free energy,

S - relative supersaturation,

S1 - a constant for B C F model,

se∩ - equilibrium concentration of growth units on surface,

β - retardation factor,

Ω - 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 S1 which determines the actual growth rate.

The BCF theory predicts that the growth rate is proportional to the

square of the supersaturation for low supersaturation, changing to a linear

dependence at higher supersaturations. The calculated growth rate for

this mechanism is found in good agreement with observations. The spiral

growth patterns have been observed on a large number of crystals grown

by different methods.

1.5 Crystal growth methods

The process of crystal growth is a controlled phase change to solid

state from solid, liquid or vapour states. Depending on the phase

transition involved, crystal growth methods are classified into four main

categories [21-25].

(1) Solid growth (solid to solid)

(2) Melt growth (liquid to solid)

(3) Vapour growth (vapour to solid)

(4) Solution growth (liquid to solid)

14 Chapter 1

Growth of crystals from liquid phase is treated as two categories

due to the independent behaviour of melt growth and solution growth

techniques. A brief description of the various methods of growth is

presented in the following sections.

1.5.1 Growth from solid

In this method, single crystals are obtained by the preferential

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

material and subsequent annealing. Large crystals of several materials,

especially metals, have been grown by this method [26]. Recently certain

rare earth compounds are also grown using this method [27]. Solid state

growth is possible by atomic diffusion also [28]. At normal temperature

such diffusion is very slow except in the case of super ionic materials.

1.5.2 Growth from melt

Melt growth is the process of crystallization by fusion and

resolidification of the pure material, crystallization from a melt on cooling

the liquid below its freezing point. In principle, all materials can be grown

in single crystal form from the melt, provided they melt congruently, do not

decompose before melting and do not undergo a phase transition

between the melting point and room temperature. The rate of growth is

much higher than any other methods and for this reason this method is

used in the growth of crystals for commercial purposes.

The melt growth can be subdivided into the following techniques:

(1) Bridgeman technique

(2) Czochralski technique

(3) Flame fusion technique

(4) Zone melting method

(5) Float zone method


In the Bridgeman technique, solidification is obtained by the

withdrawal of a boat (crucible) containing molten material through a

temperature gradient [29,30]. There are two versions – Horizontal

Bridgeman method and Vertical Bridgeman method. This method is best

suited for low melting point materials. The Bridgeman technique is used

for the growth of metals, semiconductors and alkaline earth halides and

non-linear optical crystals [31-36]. Single crystals of rare earth

compounds are also grown by this method [37].

Czochralski method is basically a crystal pulling technique used for

producing high quality crystals [38 - 41]. An advantage of this method is

that growth from a free space accommodates the volume expansion

associated with solidification of many materials. This method is widely

used to grow refractory oxides such as sapphire and ruby and

semiconductors like silicon [42 - 46]. This method has been used in the

growth of certain laser crystals as well as rare earth compounds [47 – 52].

The crystal growth of certain ferroelectric and non-linear optical material is

also reported by this method [53, 54].

The flame fusion technique (Verneuil technique) [55, 56] is a

method of crystal growth in which no crucible is used. The charge is

taken in a finely powdered form and is carried through an oxy-hydrogen

flame. There it is melted and a single crystal grows around a seed crystal.

Crystals of ruby, sapphire and corundum can be grown by this method.

In zone melting, the melt is contained in a closed container that can

be mounted horizontally or vertically. A zone or part of the solid material

is melted and this molten zone travels together with the heating elements.

An advantage of zone melting is that multiple recrystallization is possible

which permits chemical purification of the substance [57].

Float zone method is essentially zone melting method in the

vertical configuration without container. The molten zone is sustained by

16 Chapter 1

surface tension forces. This method is most suitable for materials with

high surface tension and low density. This method is used for silicon

crystals with very low oxygen concentration and also for high purity

refractory metals like tungsten and tantalum. Recently, the growth of rare

earth compounds is reported using float zone method [58].

1.5.3 Growth from vapour

Numerous methods of crystallization from the vapour phase have

been developed in connection with the requirements of modern

technology, particularly in semiconductor electronics. These methods

are used for growing bulk crystals, epitaxial films, thin coatings and

platelet crystals. Growth from vapour phase may be broadly classified

into physical vapour transport and chemical vapour transport.

Physical vapour transport involves two techniques, sublimation –

condensation and sputtering. The first method involves the sublimation

of the charge at a high temperature end of the furnace followed by the

condensation at the colder end [59]. Sputtering techniques are mainly

used to prepare thin films rather than discrete crystals. The advantage

of this method is that film growth is possible at a temperature lower than

in sublimation-condensation growth. Various crystals and epitaxial films

have been produced by PVT method [60 – 62].

Chemical vapour transport involves a chemical reaction between

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

material to be crystallized is converted into a gaseous product, which

either diffuses to the colder end or gets transported by the transporting

gas. At the colder 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 temperature of the hot zone and the crystallization zone

can be theoretically predicted [63, 64]. Commercially important materials

can be grown by this method [65-68].


1.5.4 Growth from solution

Growth of crystals from solutions is mainly a diffusion - controlled

process. The constituents of the material to be crystallized are dissolved

in a suitable solvent and crystallization occurs as the solution becomes

critically supersaturated. This may be achieved by lowering the

temperature of the solution by slow evaporation or by continuous supply

of the material to compensate for that which is precipitated out. The

solution growth method is a widely practiced method of crystal growth as

it requires lower temperature and can lead to lower density of lattice

defects.

The solution growth methods are classified according to the

temperature range and to the nature of the solvents used. The important

methods are

(a) High temperature solution growth

(b) Hydrothermal growth

(c) Low temperature solution growth

(d) Gel growth

Flux method is the most widely used high temperature solution

growth technique [69]. In this method, the components of the desired

material are dissolved in a solvent, the so called flux. Under controlled

condition, crystals are formed. Crystallization occurs at a temperature

much lower than the melting point of the crystallizing substance. Rare

earth compounds have been grown by this method [70, 71]. Another

technique in high temperature solution growth is Liquid phase epitaxy

[72] in which a thin layer of crystalline material is deposited from solution

onto a substrate of similar composition or surface structure.

In hydrothermal method, an aqueous solvent is used at elevated

temperatures and pressures to dissolve a solute which would ordinarily

18 Chapter 1

be insoluble at ambient conditions. This is an excellent method for the

growth of low temperature polymorphs of refractory materials. The

growth of rare earth based salts using this method has been reported

[73, 74].

Low temperature solution growth is sub-divided into (a) slow

cooling method and (b) slow evaporation method. In slow cooling

method, the supersaturation is achieved by a change in temperature, ie.,

the saturated solution of the material is prepared in a suitable solvent at

a high temperature and crystallization is initiated by slow cooling. In

slow evaporation method the solvent is allowed to evaporate slowly

which results in the supersaturation of the solution and crystallization of

the material.

A controlled growth of crystals with high perfection can be

achieved by gel technique [75, 76]. This technique is employed to grow

crystals with low solubility in water and high thermal instability. Metallic,

non-metallic and ferroelectric crystals are grown by this method [77, 78].

The gel method has been used for growing crystals of non-linear optical

materials [79, 80], cholesterol and urinary stones [81-83]. Most of the

oxalate crystals are insoluble in water and decompose before melting.

Hence gel method is best suited for the growth of these crystals. In the

present work, gel technique is employed for the growth of mixed rare

earth oxalate crystals.

1.6 Rare earth oxalate crystals and the present work

The electric, magnetic, luminescent and superconducting properties

of rare earth compounds make them technologically important materials

[84-86]. An alloy of Nd-Fe-B is found to have remarkable magnetic

properties [87, 88]. Some of the high temperature superconductors are

La2-x Mx CuO4 (M = Ba, Sr, Ca) and Ln Ba2 Cu3 O7-x (Ln = Y, Nd, Sm, Gd,

Dy, Ho, Er, Tm, Yb) with x = 0.2. Much of the current research, both

academic and technological, on rare earth-based material is centred on

intermetallic compounds [89].


The oxalate family of rare earths have widespread optical

applications [90, 91]. The rare earth oxalates are significant due to their

ionic conduction and because of their easy conversion into the

corresponding oxides [92, 93]. Superconducting compounds of rare

earths have been synthesized by the controlled co-precipitation of

oxalates followed by calcination [94].

The present work includes the growth and characterization of a

few mixed rare earth oxalate crystals namely gadolinium samarium

oxalate, gadolinium neodymium oxalate and gadolinium cerium oxalate.

Since these oxalate crystals decompose before melting, high

temperature methods cannot be adopted for their growth. The

insolubility of oxalates in water has been utilized for growing these

crystals by silica gel method. A detailed account of the method of

growth, characterization and study of the physical properties of the

grown crystals is given in the following chapters.

20 Chapter 1

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22 Chapter 1

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