1
Naica Cave, Mexico
0.5 My, 50 C
CaSO4ꞏ2H2O - gypsum
1.2 × 15 m
Growth of Single
Crystals
2
Growth of Single Crystals
Single crystals
- Single-crystal X-ray diffraction analysis of crystal structure
- Vital for meaningful property measurements of materials
- Measurements of anisotropic phenomena (electrical, optical,
magnetic, mechanical, thermal) in anisotropic crystals (symmetry
lower than cubic)
- Fabrication of devices
Y3Al5O12 (YAG = yttrium aluminum garnet) and beta-beryllium borate
(BBO) for doubling and tripling the frequency of CW or pulsed laser
light
SiO2 (quartz) crystal oscillators for RF generators and mass monitors
Lithium niobate for photorefractive applications
3
Thermodynamics and Kinetics of
Crystallization
As a material cools off the average kinetic energy drops
Boltzman
Distribution
T1 > T2
4
Stages of Crystallization
• Nucleation – formation of nuclei of critical size,
depositon vs. dissolution
• Growth – diffusion of material toward the critical nuclei,
crystal growth
5
Formation of Nuclei
Molecules are always bumping into each other – random
collisions - sometimes they stick – with low kinetic
energy
At lower kinetic energies more molecules stick together
= form nuclei
Cooling = lower kinetic energy
Addition of monomer
6
Transformation from Liquid to Solid
VOLUME
The energy of a crystalline phase is less than that of a liquid
The difference = the volume free energy Gv (a negative value)
As the solid grows in size, the magnitude of the total volume free
energy increases
The volume free energy Gv drives crystallization
SURFACE
When solids form in a liquid there is an interface created
The surface free energy, SL = the solid/liquid interfacial energy
associated with this interface (changed in different solvents)
As the solid grows, the total surface free energy increases (a positive
value)
The surface free energy hinders crystallization
7
Volume Free Energy
m
V
V
SRT
G
ln

ΔGV – the free energy change between the ‘monomer’ in
liquid/solution and in a unit volume of bulk crystal
S – supersaturation = the quotient of the actual concentration
[M] and the concentration of the respective species at
equilibrium with the flat crystal surface [M] (solubility),
indicates how far away from equilibrium the system is:
Vm – molar volume of the monomer composing the bulk crystal
 
 

M
M
S
 
 

M
M
S
8
Thermodynamics of Nucleation
The driving force = the supersaturated solution is not stable in energy
The total change in free energy for the nucleating system is the sum of
the two factors
For spherical nuclei
GT = 4/3  r 3 Gv + 4 r 2 SL
The volume free energy goes up
with the cube of the radius
The surface free energy goes up
with the square of the radius
GT has a maximum at a critical radius – critical free energy GN
If just a few molecules stick together, they will redissolve
If enough molecules stick together, the embryo will grow
9
r: radius of spherical nuclei
rc: critical radius
GT: total free energy change
Gs: surface free energy change
Gv: volume free energy change
GN: critical free energy change
(activation energy to nucleation)
GT = 4r2SL + 4/3r3GV
Nucleation
G
GN
4/3  r 3Gv
4 r2 SL
Surface Energy
Volume Energy
r  rc
a nucleus dissolves
rc
r > rc
a nucleus grows by itself
10
Total Free Energy of Nucleation
Surface Energy
Volume Energy
Critical
Radius
rc 4/3  r 3Gv
4 r2 SL
G

 2
3
4
3
ln4
r
V
SRTr
G
m
T 
11
Nucleation - Critical Radius rc
rc critical nuclei radius is:
S = supersaturation
SRT
V
G
r mSL
V
SL
c
ln
22 



rc = the minimum size at which a particle can survive in solution without
being redissolved
At larger supersaturation S, the critical radius of nuclei is smaller

 2
3
4
3
ln4
r
V
SRTr
G
m
T  0
)(


dr
Gd T
12
Nucleation - Critical Free Energy GN
 2
23
ln3
16
SRT
V
G mSL
N


The free energy necessary to form
stable nuclei
Thermodynamic barrier to nucleation
13
Rate of Nucleation
ΔGN – the free energy barrier to nucleation
S – supersaturation
Vm – molar volume of the bulk crystal
N – concentration of nuclei
 
  










 
 2233
23
ln3
16
expexp
SNTk
V
A
kT
G
A
dt
Nd
A
mSLN 
Arrhenius equation
The number of nuclei formed per unit time per unit volume
Nucleation Rate
14
Vm = 3.29 × 10−5 m3 mol−1 (the value for CdSe)
 
  










 
 2233
23
ln3
16
expexp
SNTk
V
A
kT
G
A
dt
Nd
A
mSLN 
15
Homogeneous Nucleation
The process of solid formation from liquid phase = homogeneous
nucleation
– random collisions of monomers and formation of nuclei
It only occurs if the material is very pure
The size of the critical radius is:
Metal crystallization from melts
T = the undercooling
Tm = melting point
Hf = the heat of fusion
Metals often experience undercooling of 50 to 500 ºC
TH
T
r
f
mSL
c


2
SRT
V
G
r mSL
V
SL
c
ln
22 



16
Heterogeneous Nucleation
Homogeneous nucleation usually only occurs under very clean
conditions
Impurities and inhomogeneities provide a “seed” for nucleation
Solidification can start on a wall
It is like cloud seeding, or water condensing on the side of a glass
Adding impurities on purpose = inoculation
GN
hetero =  GN
homo
Wetting angle
The free energy barrier to heterogeneous nucleation is always
smaller than to homogeneous nucleation
Growth
17
Growth of particle = monomer diffusion + surface reaction
The growth rate of spherical particles (dr/dt) depends on:
- the flux of the monomers to the particles (J)
- the rate of surface reaction (k)
dx
dC
DxJ 2
4
 ib CCDrJ  4
Fick: the flux J of monomers
passing through a spherical
plane with radius x
Cb = the bulk concentration of monomers within the solution
Ci = the concentration of monomers at the solid/liquid interface
Cs = the solubility of the particle
J
Cb
Ci
Cs
 si CCkrJ  2
4
The rate of surface reaction
Growth
18
Growth (dr/dt) = monomer diffusion (J) + surface reaction (k)
 sb
m
CC
r
DV
dt
dr

If diffusion is the rate
limiting factor
J
Cb
Ci
Cs
If the surface reaction is
the rate limiting factor
 sbm CCkV
dt
dr

 
k
D
r
CCDV
dt
dr sbm


the growth rate of a
particle decreases as
r increases
The growth rate
19
La Mer Mechanism
Monomer formation - concentration of monomer increases to a critical value cmin
Burst nucleation - many nuclei are generated at the same time, monomer is
consumed and its concentration drops below cmin
Growth - the nuclei grow without additional nucleation, all of the particles nucleate
simultaneously, their growth histories are the same
Control of the size distribution of the ensemble of particles during growth synthesis
of monodisperse nanocrystals
3 Separate stages:
- Monomer formation
- Burst nucleation
- Growth by diffusionCs
20
Stage I - The concentration of “monomer”, (the minimum subunit of bulk crystal)
constantly increases with time, precipitation does not occur even under
supersaturated conditions (S > 1) as the energy barrier for spontaneous
homogeneous nucleation is too high
Stage II - Nucleation occurs, the critical supersaturation (Sc) is high enough to
overcome the energy barrier for nucleation, the formation and accumulation of
stable nuclei
The rate of monomer consumption exceeds the rate of monomer supply, the
monomer concentration decreases until it reaches the level at which the
nucleation rate drops to zero
Stage III - The growth stage, nucleation stopped, the particles keep growing as
long as the solution is supersaturated by diffusion of monomer towards crystals
La Mer Mechanism
Cs
 
  sc
c
M
M
S 

21
Nucleation vs. Crystal Growth
Rate of nucleation vs. Rate of growth
Undercooling = cooling below the melting point
Relations between undercooling, nucleation
rate and growth rate of the nuclei
Ta = small undercooling, slow cooling rate
Fast growth, slow nucleation
= Few coarse crystals
Tb = larger undercooling, rapid cooling rate
Rapid nucleation, slow growth
= Many fine-grained crystals
Tc = very rapid cooling
Nearly no nucleation = glass
22
Growth of Single Crystals
Crystallization techniques: vapor, liquid, solid phase
High temperature methods
• Czochralski
• Stockbarger and Bridgman
• Verneuil
• Zone melting
Medium temperature methods
• Fluxes, Ionic Liquids
• Electrochemical from melts
• Hydrothermal
• Vapor phase transfer
• Sublimation
Low temperature methods
• Solution
• Gel
23
Jan Czochralski
(1885 – 1953)
Czochralski Method
1917
Crystal pulling technique
Single crystal growth from the melt precursor(s)
Crystal seed placed in contact with surface of melt
Temperature of melt held just above melting point =
highest viscosity, lowest vapor pressure
Seed gradually pulled out of the melt, 1 mm per hour
Melt solidifies on surface of seed
Melt and seed usually rotated counterclockwise with
respect to each other to maintain constant
temperature and to facilitate uniformity of the melt
during crystal growth, 10 rpm
Produces higher quality crystals, less defects
Inert atmosphere, often under pressure around
growing crystal and melt to prevent any materials
loss
24
Six steps in the growth of a silicon
single crystal:
a) Evacuation and heating of the
polycrystalline silicon (“pumping”)
b) Setting the temperature of the Si
melt just above 1414 ºC (“melting”)
c) Dipping the thin Si seed crystal
into the homogeneous Si melt
(“dipping”)
d) Initiating crystallization at the
neck of the thin Si seed (“necking”)
e) Adjustment of the shoulder of the
desired single crystal diameter
(“shoulder”; four positions which
portray the fourfold drawing axis
[100] are visible at the hot, light
marginal zone of the single crystal)
f) Growing phase of the single
crystal with constant diameter
(“body”)
Czochralski Method
Czochralski Method
25
Silicon
Diam 300 mm
Length 2 m
Weight 265 kg
26
Czochralski Method
The Czochralski crystal pulling technique for growing
large single crystals in the form of a rod
Subsequently cut and polished for various applications
Si Ge GaAs LiNbO3 SrTiO3 NdCa(NbO3)2
Growing bimetallic crystals like GaAs
Layer of molten inert oxide like B2O3 spread on
to the molten feed material to prevent
preferential volatilization of the more volatile
component of the bimetal
Critical for maintaining precise stoichiometry
for example Ga1+xAs and GaAs1+x which are
respectively rich in Ga and As, become p-doped
and n-doped
27
Bridgman/Stockbarger Method
Bridgman/Stockbarger method is based on a crystal growing
from the melt, involves the relative displacement of melt and a
temperature gradient furnace, fixed gradient and a moving
melt/crystal
28
Bridgman/Stockbarger Method
Gradient Bridgman/Stockbarger method is again based on
crystal growth from a melt, but now a temperature gradient
furnace is gradually cooled and crystallization begins at the
cooler end, fixed crystal and changing temperature gradient
Stockbarger and Bridgman Methods
29
Mylar0.5-1.5-189Ar (!)
Pyrex1-5434AgBr
Graphite6-601083Cu
Ir5-101790FeAl2O4
Mo2-82037Al2O3
Container materialVelocity of
grad. mm h-1
mp (oC)Crystal
Both methods are founded on the controlled solidification of
a stoichiometric melt of the material to be crystallized
Enables oriented solidification
Melt passes through a temperature gradient
Crystallization occurs at the cooler end
Both methods benefit from seed crystals and controlled
atmospheres (sealed containers)
30
Zone Melting• Crystal growth
• Purification of solids
Thermal profile furnace, RF, arc, electron beam heating
Material contained in a boat (must be inert to the melt)
Only a small region of the charge is melted at any one time
Initially part of the melt is in contact with the seed
Boat containing sample pulled at a controlled velocity through
the thermal profile furnace - zone of material melted
Oriented solidification of crystal occurs on the seed
31
Zone Melting
• Crystal growth
• Purification of solids
Zone refining methods for
purifying solids
Partitioning of impurities occurs
between melt and the crystal
Impurities concentrate in liquid
more than the solid phase, swept
out of crystal by moving the
liquid zone
Used for purifying materials like
W, Si, Ge to ppb level of
impurities, often required for
device applications
32
A small slice of the sample is molten
and moved continuously along the
sample
Impurities normally dissolve
preferably in the melt
Segregation coefficient k:
k = csolid/cliquid
(c: concentration of an impurity)
Only impurities with k < 1 can be
removed by zone melting !!
Zone Melting
Verneuil Fusion Flame Method
33
1902 - French chemist Auguste Verneuil
The first commercially successful method of manufacturing
synthetic gemstones - ruby, sapphire, diamond simulants
rutile and strontium titanate
Verneuil Fusion Flame Method
34
Lowered 10 mm/hour
Useful for growing crystals
of extremely high melting metal oxides
Examples include:
• Ruby from Cr3+/Al2O3 powder
• Sapphire from Cr2
6+/Al2O3 powder
• Spinel, CoO, ferrites
Starting material fine powder
Passed through O2/H2 flame
or plasma torch
Melting of the powder occurs in the
flame
Molten droplets fall onto the surface of
a seed or growing crystal
Controlled crystal growth
Flux Method
35
Material dissolved in a suitable flux = solvent (metals, fluorides, oxides),
lower melting point than the pure solute
Single crystals grown from supersaturated solution
Suitable method for materials which:
- vaporize or dissociate at temperatures above their mp
- there are no suitable containers at elevated temperatures
Material Flux
As Ga
B Pt
Si, Ge Pb, Zn, Sn
GaAs, GaP Pb, Zn, Sn
BaTiO3 KF
ZnO PbF2
ZnS SnF2
MgFe2O4 NaF
Co3O4 B2O3 – PbO
Fe2O3 Na2B4O7
TiO2 Na2B4O7 – B2O3
AlF3
2.0 g of AlF3, 25.0 g of PbCl2, 2.5 g PbF2
24 h at 1200 K, cooled at 4 deg h1
down to 723 K
Thick platelets and small cubes
Solution Methods
36
Suitable for materials with a reasonable solubility in the selected solvent:
water, organic solvents, NH3(l) , HF, SO2(l)
Nucleation – homogeneous – heterogeneous
Dilute solution, solvent with low solubility for given solute
Supersaturated solution, seed crystals
Single crystals grown at constant supersaturation
Techniques:
* slow evaporation
* slow cooling
* vapor diffusion
* antisolvent diffusion
* reactant diffusion
* recirculation, thermal differential, convection
* cocrystallants (OPPh3 for organic proton donors)
* counterion, similar size of cation and anion least soluble
* ionization of neutral compounds, protonation/deprotonation, H-bonding
37
KDP crystals (KH2PO4)
• grown from supersaturated
solution
• crystal seed
• slow cooling
a frequency converter - converts
the infrared light at 1053 nm into
the ultraviolet at 351 nm
Hydrothermal Crystallization/Synthesis
38
1957 - Bell Labs
Autoclave with Water medium
High temperature growth, above normal boiling point
Water acts as a pressure transmitting agent
Water functions as solubilizing phase
Often a mineralizing agent is added to assist with the
transport of reactants and crystal growth
Speeds up chemical reactions between solids
Crystal growth hydrothermally involves:
• Temperature gradient reactor = autoclave
• Dissolution of reactants at one end
• Transport with help of mineralizer to seed at the
other end
• Crystallization at the other end
Hydrothermal Crystallization/Synthesis
39
Useful technique for the synthesis and crystal growth of phases
- unstable in a high temperature preparation in the absence of water
- materials with low solubility in water below 100 oC
Hydrothermal growth of quartz crystals
Annual global production hundreds of tons of quartz crystals
Uses of single crystal quartz: Radar, sonar, piezoelectric transducers,
monochromators, XRD
Water medium, nutrients 400 oC, seed 360 oC, pressure 1.7 kbar
Mineralizer 1M NaOH
A mineralizer is a complexing agent (not too stable) for the
reactants/precursors that need to be solublized (not too much) and
transported to the growing crystal
Role of the mineralizer - control of crystal growth rate
Choice of mineralizer, temperature and pressure
Solubility of quartz in water is important
Hydrothermal Quartz Synthesis
40
Hydrothermal growth of quartz crystals: SiO2 + 2 H2O ⇄ Si(OH)4
0.3 wt% even at supercritical temperatures >374 oC
Need mineralizing reactions:
NaOH mineralizer, dissolving reaction, 1.3-2.0 kbar
3 SiO2 + 6 OH- ⇄ Si3O9
6- + 3 H2O
Na2CO3 mineralizer, dissolving reaction, 0.7-1.3 kbar
CO3
2- + H2O ⇄ HCO3
- + OHSiO2
+ 2 OH- ⇄ SiO3
2- + H2O
NaOH creates growth rates about 2x greater than with Na2CO3 because of
different concentrations of hydroxide mineralizer
Hydrothermal Synthesis
41
Some materials have negative solubility coefficients, crystals grow at the
hotter end in a temperature gradient hydrothermal reactor
Example: -AlPO4 (Berlinite) - important for its high piezoelectric
coefficient (larger than -quartz with which it is isoelectronic) used as a
high frequency oscillator
Powdered AlPO4 cool end of reactor - negative solubility coefficient!!!
H3PO4/H2O mineralizer, AlPO4 seed crystal at hot end
Emeralds Cr3+/Be3Al2Si6O18
SiO2 powder at hot end 600 oC, NH4Cl
or HCl/H2O mineralizer, 0.7-1.4 kbar,
cool central region for seed, 500 oC,
Al2O3/BeO/Cr3+ dopant powder mixture
at other hot end 600 oC
6 SiO2 + Al2O3 + 3 BeO ⇄ Be3Al2Si6O18
Beryl contains Si6O18
12- six rings
Hydrothermal Synthesis
42
Metal crystals
Metal powder at cool end 480 C, Mineralizer 10M HI / I2
Metal seed at hot end 500 C
Dissolving reaction that also transports Au to the seed crystal:
Au + 3/2 I2 + I- ⇄ AuI4
Metal
crystals grown this way include: Au, Ag, Pt, Co, Ni, Tl, As
Diamonds
Ni + C + H2O ⇄ diamond
Carbon films on SiC fibers
SiC + 2 H2O ⇄ C + 2 H2 + SiO2 100 MPa, 300- 600 oC
Zeolites
Al(OH)3, SiO2, NaOH, template  Mx/n [(AlO2)x(SiO2)y]. mH2O
Hydrothermal Synthesis
43
Pressure, volume, temperature tables
Critical point of water: 374.2 oC, 218.3 bar
Density of liquid water decreases with T
Density of water vapor increases with T
Density of gas and liquid water the same
0.32 g cm-3, at the critical point
Tables of pressure versus temperature for different initial volume filling of
autoclave must be consulted to establish a particular set of reaction conditions for
a hydrothermal synthesis or crystallization
Safety: an autoclave explosion
Liquid level in autoclave rises for > 32% volume filling
For 32% volume filling liquid level remains unchanged and becomes fluid
at critical temperature
volume
filling %
Bulk-Material Dissolution Technique
44
Large zeolite crystals: up to 3 mm, SOD, MFI, ANA, CAN, JBW
Autoclave, PTFE liner
quartz tube (SiO2) TPAOH, HF, H2O 200 C, 25-50 days
ceramic tube (SiO2, Al2O3) NaOH, H2O 100-200 C, 7-20 days
Small surface area, low dissolution rate, saturation concentration
maintained, only a few nuclei are produced at the beginning, no large
crystals formed in the stirred reactions, concentration gradients
Decomplexation Crystallization
45
Crystallization under ambient conditions, low temperature and
pressure, provides kinetic products, control of crystal size and
morphology, habit
AgX, X = Cl, Br, I
MX, M = H, Na, K, NH4
AgI + HI ⇄ H+ + [AgI2]- aqueous solution
Overlayer absolute ethanol, HI diffusion, decomplexation of AgI,
hexagonal plates 5 mm
AgX + 2 NH3 ⇄ [Ag(NH3)2]+ + XX
= Cl, Br, slow evaporation (3-5 days), AgX crystals
Decomplexation Crystallization
46
AgI + KI ⇄ K+ + [AgI2]Concentration
gives K[AgI2] crystals
Dilution by slow diffusion gives 20 mm AgI crystals
Warming gives AgI crystals
Inverse temperature dependence of AgI solubility in KI
CuCl + HCl ⇄ H+ + [CuCl2]HgI2
+ KI ⇄ [HgI3]- ⇄ [HgI4]2PbO
+ hot KOH solution
slow cooling provides PbO as 2 mm yellow needles and
1 mm red blocks
Complexation-Mediated Crystallization
47
Salts with high lattice energy: fluorides, carbonates, acetates
Solubilized in organic solvents by crown ethers
Crystallization provides uncomplexed salts
NaOOCCH3.3H2O dissolves in cyclohexane with 15-crown-5
prismatic crystals
Complexation-Mediated Crystallization
48
Two soluble salts react to produce an insoluble phase
- aqueous solutions
- nonaqueous solvents
CaCO3 calcite TD stable phase at room temp., in H2O
vaterite kinetic product
aragonite TD stable at high temperature
CaCl2 (in MeOH) + NaHCO3 (in MeOH, 18-crown-6)
Microcrystalline calcite
upon aging converts to nanocrystalline vaterite, surface stabilization by
surface chelatation
The Gel Method
49
Large single crystals
Hydrogels: silicagel (water glass), polyvinyl alcohol,
gelatin, agar
Silicate gel
Impregnation with metal or ligand, setting the gel = condensation,
crosslinking, pH control of the condensation rate
Layered with the solution of ligand or metal
Slow diffusion, crystal growth
CuSO4 + [NH3OH]Cl  Cu
Pb(OAc)2 + Zn  Pb + Zn(OAc)2
Pb(OAc)2 + KI  PbI2 + 2 KOAc
Liesegang rings, agates
RbSnBr3, CsSb2I5 semiconductors
The Gel Method
50
Nonaqueous gels
PEO (MW = 100 000) in 1,2-dichloroethane + MeOH,
EtOH, PrOH, DMF, CH3CN, DMSO
Impregnation with metal or ligand
Layered with the solution of ligand or metal
Slow diffusion, crystal growth
U-tube, counter-diffusion A + B  C
Concentration programming, increasing
concentrations
Ostwald rippening = larger xtals grow, smaller dissolve
Electrochemical Reductive Crystal Growth
51
Molten mixtures of precursors, product crystallizes from melt
Melt electrochemistry: Electrochemical reduction
CaTi(IV)O3 (perovskite) / CaCl2 (850 oC)  CaTi(III)2O4 (spinel)
Na2Mo(VI)O4 / Mo(VI)O3 (675 oC)  Mo(IV)O2 (large crystals)
Li2B4O7 / LiF / Ta(V)2O5 (950 oC)  Ta(II)B2
Na2B4O7 / NaF / V(V)2O5 / Fe(III)2O3 (850 oC)  Fe(II)V(III)2O4 (spinel)
Na2CrO4 / Na2SiF6  Cr3Si
Na2Ge2O5 / NaF / NiO  Ni2Ge
Phosphates  phosphides
Carbonates  carbides
Sulfates  sulfides
Inverse Temperature Crystallization
52
methylammonium lead tribromide
(CH3NH3PbBr3)
high-energy radiation detectors
MAPbX3 perovskites exhibit inverse
temperature solubility behavior in
certain solvents
The solubility of MAPbBr3 in DMF
0.80 ± 0.05 g ml-1 at r.t
0.30 ± 0.05 g ml-1 at 80 C
Oriented Crystal-Crystal Intergrowth
53
methylammonium lead tribromide
(CH3NH3PbBr3)
high-energy radiation detectors
Inverse temperature crystallization
MAPbX3 perovskites exhibit inverse
temperature solubility behavior in
certain solvents
54
Synthesis of Amorphous Materials
Quenching of molten mixture of metal oxides with a glass former
(P2O5, V2O5, Bi2O3, SiO2, CaO, …), large cooling rates required (>107 K s1)
Ion beam sputtering
Thermal evaporation
Thermal decomposition of organometallic precursors (Fe(CO)5, …)
Amorphous Cr2O3, MnO2, PbO2, V2O5, Fe2O3
Sonochemical decomposition of organometallic precursors
Fe(CO)5, M(acac)n,…
Precipitation on metal hydroxides, transformation to hydrous oxides
MW heating of metal salt solution
Amorphous Cr2O3, Fe2O3