of
Thermodynamics crystal Growth
for
of
Thierry DUFFAR
Professor at Grenoble Institute of Technology
SIMAP Laboratory
France
[email protected]
15th International Summer School on Crytsal Growth – ISSCG-15
Outline
Introduction: thermodynamics OF crystal growth
Thermodynamics FOR crystal growth
1) Minimization of energy
the equilibrium shape of a crystal
Chemical reactions
2) Equilibrium
Phase diagrams
Point defects
3) Out of equilibrium
Driving force for phase change
4) Application
Thermodynamics of epitaxy
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Josiah Willard Gibbs
1839-1903, New Haven, Connecticut
Combination of 1st and 2nd principles of thermodynamics
Introduction of Enthalpy and « Gibbs » free energy
Chemical potential
Multi-phase systems
Variance, phase rule
Nucleation theory
Gibbs-Thomson equation (curved surfaces)
Statistical physics: petit-, grand- and micro-canonical ensembles
And many other things in mathematics (vector analysis)
and physics (optics)
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Introduction
Chemistry (species mixing)
dU = TdS − PdV + ∑ µi dni
i
δQ, Heat
Mechanical Intensive Extensive δW
Work
Variable variable
Force
F (N)
l (m)
Fdl
Pressure
P (Nm-2)
V (m3)
-PdV
Elastic
St (Nm-2)
εV (m3)
-StεdV
Surface
σ (N m-1)
S (m2)
-γdS
Electric
E (V)
q (Cb)
Edq
Magnetic
HB (Nm-2)
V (m3)
HBdV
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Introduction
There are several thermodynamic functions:
Enthalpy:
H = U + PV
dH = TdS + VdP + ∑ µi dni
i
Helmotz free energy:
F = U − TS
dF = − SdT − PdV + ∑ µi dni
Gibbs free energy:
G = H − TS
i
dG = − SdT + VdP + ∑ µi dni
i
Most crystal growth processes are under constant T and P: dG=0
Gibbs energy is generally the most convenient
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Introduction
Thermodynamics OF crystal growth
Crystal growth deals with the production of a crystal (solid) from
another phase (liquid, gas …).
When the crystal is in equilibrium with the fluid at the temperature TE:
∆G = ∆H − TE ∆S = 0
Or:
∆S =
∆H
Energy of the transformation
(bonding)t
TE
Disorder introduced by the transformation
Crystal Growth is changing disorder into ordered bonding
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Exercise
Thermodynamics OF crystal growth
What is the entropy associated to:
-Growth of Silicon from the melt by the Czochralski
method
Tmelt=1683K, ΔHmelt=50.7 103 J.mol-1
ΔS= 30 J.mol-1.K-1
-
Growth of SiC from the vapor by the Lely method
Tprocess=1683 K, ΔHsublimation= 1233 103 J.mol-1
ΔS= 731 J.mol-1.K-1
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Nomenclature
a activity
e thickness
k Bolzmann constant
n mole number
r radius
x molar fraction
<…> solid
A area
E energy, or electric field
D diffusion coefficient
F Helmotz free energy
G Gibbs free energy
H enthalpy
J flux
K equilibrium constant
N number of … (bonds, atoms, layers..)
P pressure
Q heat
R perfect gas constant
S entropy
T temperature
U internal energy
V volume
W work (mechanical energy)
(…) liquid
((…)) liquid solution
15th International Summer School on Crystal Growth – ISSCG-15
γ activity coefficient
δ lattice parameter
ε electrical permittivity
λ interaction parameter
μ Chemical potential
σ interfacial energy
ω number of configurations
Ω molar volume
[…] gas…
LAST NAME, First Name – talk id
1) Minimization of energy
Equilibrium shape of a crystal
Good use of thermodynamics for chemical reaction studies
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Minimization of energy: Wulff plot
dG = − SdT + VdP + σdA + ∑ µi dni
i
What is the shape of a crystal in equilibrium with its surrounding?
Wulff theorem:
dF = ∫ σ ( hkl )dA( hkl )
should be minimal
The surface energy σ is function of orientation <hkl>:
Si-vapor:
σ(100)=2.13 J.m-2
σ(110)=1.51 J.m-2
σ(111)=1.23 J.m-2
R.J. Jaccodine, J. Electrochem. Soc. 110 (1963) 524-527.
R. Drosd, J. Washburn, J. appl. Phys. 53 (1982) 397-403.
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Minimization of energy: Wulff plot
Equivalent to:
“distances of faces to the center are proportional to their surface energy”
Wulff plot and construction:
σ plot
σ001
σ111
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Minimization of energy: Wulff plot
Exercise
What is the equilibrium shape of the crystal with this Wulff plot?
σ(θ)
σ plot
R.F. Sekerka p. 57 in “Crystal growth-from Fundamentals to Technology”, Müller G., Métois J.J., Eds. (2004) Elsevier
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Minimization of energy: Wulff plot
Don’t confuse the EQUILIBRIUM shape and the KINETIC shape
(the survival faces are the slowest).
(111)
slowest
(100)
slowest
15th International Summer School on Crystal Growth – ISSCG-15
(110)
slowest
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Minimization of energy: chemical reactions
Thermochemistry allows computing the possibility of a chemical
reaction: equilibrium is obtained when the Gibbs energy of the
system is minimal (constant T and P).
CORRECT use of Gibbs energy is mandatory
Example: is it possible to melt Si in a sapphire (or possibly sintered
alumina) crucible without pollution?
Si droplets after
solidification on
sapphire
silica
substrates
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Minimization of energy: chemical reactions
Common, but insufficient treatment:
simple use of Ellingham diagram
∆G Al → Al O < ∆GSi → SiO
2
3
2
This means that alumina is more stable
than silica
It does NOT mean that Si cannot react
with Al2O3!
http://www.doitpoms.ac.uk/tlplib/ellingham_diagrams/interactive.php
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Minimization of energy: chemical reactions
1
3
Al2O3 ↔ (( Al ))Si + (( O ))Si
2
2
K Al = aAl a
3/ 2
O
=e
∆G0Al O
aAl = γ (( Al ))Si xAl ≈ γ
aO = γ (( O ))Si xO ≈ γ
2 3
∆G 0
2 RT
Al2O3
∞
(( Al ))Si
xAl
∞
(( O ))Si O
= −1128 10 3 J .mol −1 ( 1700 K )
γ ((∞ Al )) = 0.42
Si
γ ((∞O )) = 8 10 −6
Si
x
3xAl = 2 xO
CONCLUSIONS
10-4
xAl=1.33
xO= 2 10-4
1)133 ppm Al pollution of Si melt
2) xO > O solubility limit in Si:
SiO2 is formed!
3) Taking a=x gives xO= 1.36 10-7
Wetting of ceramics by molten silicon and silicon alloys: a review B. Drevet. N. Eustathopoulos, J. Mat. Sci. 47 (2012) 8247-8260.
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
2) Equilibrium: phase diagrams
Thermochemistry of solutions
Phase equilibrium
Manipulation of phase diagram
Point defects
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Equilibrium: phase diagrams
Thermochemistry of solutions
Gibbs energy of ONE phase with two components A and B
id
xs
G α = G 0 + ∆Gmix
+ ∆Gmix
Before mixing
Ideal solution
Excess energy
id
id
id
id
∆ G mix
= ∆ H mix
− T ∆ S mix
= −T ∆ S mix
∆S
id
mix
= k ln ω
(
n A + n B )!
ω=
n A ! nB !
id
∆ G mix
= RT ( x αA ln x αA + x αB ln x αB )
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Equilibrium: phase diagrams
Thermochemistry of solutions
REGULAR model of interactive mixing (liquid solution)
∆G
xs
mix
= ∆H
xs
mix
E AA + E BB
= N AB ( E AB −
) = λx A x B
2
ln γ B =
λ
RT
( 1 − x B )2
DLP (δ Lattice Parameter) model (solid solution ex: GaxAl(1-x)As)
ln γ B =
λ
RT
( 1 − x BC )2
2  δ AC + δ BC 
λ = 5.10 7 (δ AC − δ BC ) 

2
− 4 .5


G.B. Stringfellow J. Crystal Growth 27 (1974) 21.
Many mixing energetical models exist, which all end with γ
function of various interaction parameters. In all cases:
µ A = µ A0 + RT ln γ A x A
µ B = µ B0 + RT ln γ B xB
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Equilibrium: phase diagrams
Thermochemistry of solutions
λ =0
λ < 0 A − B attraction
λ > 0 A − B repulsion
15th International Summer School on Crystal Growth – ISSCG-15
λ >> 0
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Equilibrium: phase diagrams
Equilibrium between phases α, β …: they don’t exchange energy,
they are at the same energy level.
G α = G β = ...
dG = d ( G α + G β + ...) = 0
Constant T and P:
α
α
β
β
µ
dn
+
µ
dn
∑ i i ∑ i i + ... = 0
i
α
i
β
µ = µi = ...
i
α

G
∂
µ αi = 
 ∂ni


 P ,T ,n j
Each chemical component i should be in equilibrium between the
various phases
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Equilibrium: phase diagrams
In case of strong
repulsion (λ>>0)
Two phases appear
(demixion)
α1
α2
µB = µB
 ∂G α
µ B = 
 ∂nB
α


 P ,T ,n A
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Equilibrium: phase diagrams
Construction of a
3 solid phase
binary phase
diagram
λ<<0
strong attraction: α+β
C
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Equilibrium: phase diagrams
Manipulation of phase diagram
Langasite
La3Ga5SiO14
µ A = µ A0 + RT ln γ A x A
500V.cm-1
Congruent growth
from the melt!
∂
µ A = µ + RT ln γ A x A +
∂x A
0
A
1

 2 Ω AεE 
S.Uda, X. Huang, S. Koh J. of Crystal Growth 281 (2005) 481–491
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Equilibrium: point defects
Intrinsic
• Empty site : Vacancy, VSi
• Misplaced atom : Interstitial (SiI)
• Vacancy + interstitial = Frenkel Defect
Extrinsic
• Foreign atom: Impurity (Insertion or Substitution)
Substitutional
impurity
Interstitial
Impurity in
Insertion
Vacancy
Frenkel defect
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Equilibrium: point defects
Vacancy
4 broken bonds
2 created bonds
Interstitial
4 created bonds + lattice distorsion
EV ~ Ecoh /2
EI high
EV = 0.7 to 1 eV (cf. Ecoh)
EI = 3 to 5 eV (strong distorsions)
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Equilibrium: point defects
Point defects are thermally activated: thermodynamic defects
N0 atoms, NV vacancies
.
.
.
.
.
. .
.
.
.
EV = 1 eV
N0 = 6 1023 mol-1
∆G = NV EV + ∆H Mix – T ∆S Mix
Ideal solution:
∆H Mix = 0
Boltzmann dist.:
∆S Mix = k
ln( N 0 + NV )!
N 0 ! NV !
N
d∆G
= 0 = EV – kT ln ( 0 )
dNV
NV
→
NV = N 0 e
-
EV
kT
1000 K : 6 1018 mol-1
300 K : 6 107 mol-1
In fact recombination, at Tmelt : [V]Si ≈1014 to 1015 mol-1
[I]Si≈ 0
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Vacancy concentration
Equilibrium: point defects
Temperature, K
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Equilibrium: point defects
µ-void precipitation
E1 = N.EV (-TΔSMix)
E2 = σsv4πR2
N < Ncrit = 9/16. π a6 (σsv/EV)3 < N
4 3
πR = NVV
3
15th International Summer School on Crystal Growth – ISSCG-15
for Si: VV = Vatom
a3
=
8
a
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Equilibrium: point defects
µ-voids and interstitial precipitation
Vacancies
Interstitials
106 cm-3
1010 cm-3
Annealed under Ar, H2, > 1000°C
As in GaAs, very stable
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Equilibrium: point defects
Exercise
What is the density of VGa in GaAs at 1200°C?
Will µ-voids precipitate?
If yes, what will be their density, their diameter, their mean distance?
Surface energy: 1,2 J.m-2
EV= 0,86 eV
lattice parameter a=0,56 nm
the crystallographic structure is the same than for Si, one atom over two
being Ga and the other one As
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Equilibrium: point defects
Intrinsic Point defects in a compound: example of GaAs
GaAs non-congruent
-Excess of Ga or As
- VGa, VAs, AsGa, GaAs
-Precipitation of Ga, As or
Vac.
-Anionic and cationic Frenkel
-Shottky (VGa+VAs)
Wenzl H., Oates W.A., Mika K., in: Handbook of Crystal Growth, Vol 1a, Hurle D.T.J. (ed.) North Holland, Amsterdam, 1993
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Equilibrium: point defects
Intrinsic Point defects in a compound: example of GaAs
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Equilibrium: point defects
Intrinsic Point defects in a compound: example of GaAs
Striations => stoichiometry=> dislocations
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Equilibrium: point defects
Intrinsic Point defects in a compound: example of GaAs
E
EL2 = As in a Ga site
CB
Important defect:
fixes the Semi Insulating character of GaAs
(109cm-3 vs. 1015cm-3 C)
EC
Gap
1.43 eV
++
As 0As + VGa− ↔ AsGa
+ VAs+ + 4e −
xEL2 = x As ++ = K
Ga
xV −
Ga
EL2
Ev
VB
xV + N 4
As
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Equilibrium: point defects
Intrinsic Point defects in a compound: example of GaAs
As pressure control
Lagowski J., Gatos H.C., Aoyama T., LIN D.G., Appl. Phys. Lett. 45 (1984) 680-682.
.
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
3) Out of equilibrium
Driving force and supersaturation
Various Crystal Growth processes
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Out of equilibrium
No crystal growth at equilibrium!
μ
Liquid
Δμ
Crystal
μLiquid=μCrystal=
μ0
ΔT
TMelt
T
Δμ = is the driving force for crystallization
ΔT is the undercooling
Growth rate depends directly on Δμ
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Out of equilibrium
Growth from the melt
∆µ = µ Liquid - µCrystal = (µ Liquid - µ0 ) − (µCrystal - µ0 ) =
T
∫
∂µ Liquid
∂T
TMelt
∆µ =
TMelt
∫ ∆S Melt dT ≈ ∆S Melt ( Tmelt − T ) = ∆S Melt ∆T = ∆µ =
T
∂µCrystal
T
∫
dT -
TMelt
∆H Melt
Tmelt
∆T
∂T
dT
Undercooling
Growth from vapour
∆µ = µVapor - µCrystal = ∆µ =
P
∫
P0
∂µ Liquid
∂P
P
dP - ∫
∂µCrystal
P0
Growth from solution
15th International Summer School on Crystal Growth – ISSCG-15
∂P
P
dP = ∫ ( v vapor -vCrystal )dP
P0
∆µ = kT ln
P
P0
∆µ = kT ln
x
x0
Supersaturation
Supersaturation
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
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Out of equilibrium
Mother phase
e
Growth by 2D nucleation
∆GNucleus = -πr 2 e
∂∆GNucleus
=0
∂r
Crystal
∆µ
+ 2πreσ
Ω
σΩ
r* =
∆µ
∆G
*
Nucleus
πΩσ 2 e
=
∆µ
∆T
T0
ΔG
or
∆P
P0
Example of Si
∆G*Nucleus
r*
r
P. Rudolph, in: Crystal growth Technology, H.J. Scheel and T. fukuda (eds.) Wiley (2003).
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
4) Application to epitaxy
“Far from equilibrium”?
Driving force controls kinetics
Stoichiometry and composition of layers: physical properties
Layer structure
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Application to epitaxy
∆T
T0
or
∆P
P0
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Application to epitaxy
Example of GaAs
[AsH 3 ] + [( CH 3 )3 Ga] ↔
GaAs
1
[As4 ] + [Ga] ↔ GaAs
4
a GaAs
K=
PEq [Ga ]PEq1 /[4As4 ]
P[Ga ]P[1As/ 44 ]
1
∆µ = µ[Ga ] + µ[ As4 ] − µ GaAs = RT ln
4
PEq [Ga ]PEq1 /[4As4 ]
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Application to epitaxy
Δμ*
Example of GaAs
kJ.mol-1
300
200
100
0
LPE
VPE OMCVD MBE
“Far from equilibrium”
Interface kinetics >> diffusion
Stringfellow G.B. in “Crystal growth-from Fundamentals to Technology”, Müller G., Métois J.J., Eds. (2004) Elsevier
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Application to epitaxy
Generally thermodynamic equilibrium applies at the interface
P[Ga]P[1As/ 44 ] >> PEq[Ga]PEq1/[4As4 ]
The process runs with a strong excess of As: P[ As4 ] is constant:
PEq[Ga] << P[Ga] << 4P[ As4 ]
1) The process is controlled by the diffusion of Ga: J = DGa
2) Layer stoichiometry has nothing to do with
P[Ga]
(P[
Ga]
− PEq[Ga]
RT
P[ As4 ]
3) Changing the flow rate of [Ga] has no effect on layer stoichiometry
4) Example of GaxAl(1-x)As: x is totally controlled by
15th International Summer School on Crystal Growth – ISSCG-15
P[Ga]
P[ Al]
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
)
Application to epitaxy
Manipulation of phase diagram
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Application to epitaxy
layer structure
Gibbs energy per unit area
G
n
σ Substrate
atomic layers
Substrate
σ Substrate / layer
+ σ layer / vapor
n
Bu
lk
de
po
µ n −layer =
>
sit
µbulk
15th International Summer School on Crystal Growth – ISSCG-15
∂G
∂n
Stable
uniform layer
∂G
=
∂n
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Application to epitaxy
layer structure
Gibbs energy per unit area
G
σ Substrate / layer
lk
Bu
+ σ layer / vapor
si t
po
de
Substrate
γ Substrate
n
∂G
µbulk =
∂n
>
µ n −layer
15th International Summer School on Crystal Growth – ISSCG-15
∂G
=
∂n
Unstable
layer
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Application to epitaxy
layer structure
Gibbs energy per unit area
G
σ Substrate / layer
lk
Bu
+ σ layer / vapor
nc
si t
po
de
Substrate
nc
σ Substrate
n
µbulk > µ n −layer
µbulk < µ n −layer
Stable
thin layer
for n<nc
Unstable
layer for n>nc
15th International Summer School on Crystal Growth – ISSCG-15
DUFFAR
Thierry
- Thermodynamics
LAST
NAME,
First
Name – talk id
Conclusion
Transport
Kinetics
How atoms are carried to the interface
How atoms are piled up at the interface
Thermodynamics apply
because kinetic>>growth rate
15th International Summer School on Crystal Growth – ISSCG-15
LAST NAME, First Name – talk id
Conclusion
Thermodynamic is a powerful tool:
-Is it is possible to grow a crystal?
-By which technique?
-Growth parameters?
-Crystal composition?
-Crystal thermodynamic defects?
-Crystal quality (stable or not)?
15th International Summer School on Crystal Growth – ISSCG-15
LAST NAME, First Name – talk id
Further reading
I.V. Markov „Crystal Growth for beginners“
World Scientific (2003, 2nd edition)
ISBN-13 978-981-238-245-0
W. Kurz, D.J. Fischer „Fundamentals of solidification“
Transtech Publication Ltd (1998, 4th edition)
ISBN 0-87849-804-4
D.T.J. Hurle, editor „Handbook of Crystal Growth“, Vol.
1-a North Holland (1993)
ISBN 0-444-88908 6
15th International Summer School on Crystal Growth – ISSCG-15
LAST NAME, First Name – talk id