Duffar lecture ISSCG-15 - Politechnika Gdańska · Equilibrium: phase diagrams Manipulation of...

15th International Summer School on Crytsal Growth – ISSC G-15

Thermodynamics crystal Growth

Thierry DUFFARProfessor at Grenoble Institute of Technology

SIMAP LaboratoryFrance

[email protected]

of

offor

15th International Summer School on Crystal Growth – ISSC G-15 LAST NAME, First Name – talk id

Outline

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

Thermodynamics FOR crystal growth

Introduction: thermodynamics OF crystal growth

DUFFAR Thierry - Thermodynamics


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)



Introduction

ii

idnPdVTdSdU ∑+−= µ

δQ, Heat

Chemistry (species mixing)

MechanicalWork

Intensive Variable

Extensive variable

δδδδW

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



Introduction

PVUH +=

There are several thermodynamic functions:

Enthalpy:

Helmotz free energy:

Gibbs free energy:

ii

i

ii

i

ii

i

dnVdPSdTdG

dnPdVSdTdF

dnVdPTdSdH

∑

∑

∑

++−=

+−−=

++=

µ

µ

µ

TSUF −=

TSHG −=

Most crystal growth processes are under constant T and P: dG=0

Gibbs energy is generally the most convenientDUFFAR Thierry - Thermodynamics


Introduction

Thermodynamics OF crystal growth

Crystal growth deals with the production of a crystal (solid) fromanother phase (liquid, gas …).

When the crystal is in equilibrium with the fluid at the temperature TE:

Or:

0STHG E =−= ∆∆∆

ET

HS

∆∆ =Energy of the transformation(bonding)t

Disorder introduced by the transformation

Crystal Growth is changing disorder into ordered bonding



Exercise

Thermodynamics OF crystal growth

What is the entropy associated to:

-Growth of Silicon from the melt by the Czochralskimethod

Tmelt=1683K, ΔHmelt=50.7 103 J.mol-1

- Growth of SiC from the vapor by the Lely methodTprocess=1683 K, ΔHsublimation= 1233 103 J.mol-1

ΔS= 30 J.mol-1.K-1

ΔS= 731 J.mol-1.K-1



Nomenclature a activity

e thicknessk Bolzmann constant

n mole number

r radius

x molar fraction

γ activity coefficient

δ lattice parameterε electrical permittivity

λ interaction parameterμ Chemical potentialσ interfacial energy

ω number of configurationsΩ molar volume

A area

E energy, or electric fieldD diffusion coefficient

F Helmotz free energyG Gibbs free energyH enthalpy

J fluxK equilibrium constant

N number of … (bonds, atoms, layers..)P pressureQ heat

Rperfect gas constantSentropy

T temperatureU internal energyV volumeWwork (mechanical energy)

<…> solid (…) liquid ((…)) liquid solution […] gas…


1) Minimization of energy


Equilibrium shape of a crystal

Good use of thermodynamics for chemical reaction studies


Minimization of energy: Wulff plot

ii

idndAVdPSdTdG ∑+++−= µσ

What is the shape of a crystal in equilibrium with its surrounding?

∫= )hkl(dA)hkl(dF σWulff theorem: 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.




Equivalent to: “distances of faces to the center are proportional to their surface energy”

Wulff plot and construction:


σ plot

σ111

σ001


Minimization of energy: Wulff plotExercise

What is the equilibrium shape of the crystal with this Wulff plot?

R.F. Sekerka p. 57 in “Crystal growth-from Fundamentals to Technology”, Müller G., Métois J.J., Eds. (2004) Elsevier


σ(θ)

σ plot



Don’t confuse the EQUILIBRIUM shape and the KINETIC shape (the survival faces are the slowest).

(111) slowest

(110) slowest

(100) slowest



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 silicasubstrates




Common, but insufficient treatment:

simple use of Ellingham diagram

http://www.doitpoms.ac.uk/tlplib/ellingham_diagrams/interactive.php

232 SiOSiOAlAl GG →→ < ∆∆

This means that alumina is more stable than silica

It does NOT mean that Si cannot react with Al2O3!




SiSi32 ))O((2

3))Al((OAl

2

1 +↔

Wetting of ceramics by molten silicon and silicon alloys: a review B. Drevet. N. Eustathopoulos, J. Mat. Sci. 47 (2012) 8247-8260.

OAl

O))O((O))O((O

Al))Al((Al))Al((Al

RT2

G

2/3OAlAl

x2x3

xxa

xxa

eaaK

SiSi

SiSi

0

3O2Al

=

≈=

≈=

==

∞

∞

γγ

γγ

∆

6))O((

))Al((

130

108

42.0

)K1700(mol.J101128G

Si

Si

3O2Al

−∞

∞

−

=

=

−=

γ

γ

∆

xAl=1.33 10-4

xO= 2 10-4

CONCLUSIONS

1)133 ppm Al pollution of Si melt2) xO > O solubility limit in Si:

SiO2 is formed!3) Taking a=x gives xO= 1.36 10-7



2) Equilibrium: phase diagrams


Thermochemistry of solutions

Phase equilibrium

Manipulation of phase diagram

Point defects

15th International Summer School on Crystal Growth – ISSC G-15 LAST NAME, First Name – talk idDUFFAR Thierry - Thermodynamics

Gibbs energy of ONE phase with two components A and B

xsmix

idmix

0 GGGG ∆∆α ++=

Before mixing Ideal solution Excess energy

( )!n!n

!nnlnkS

STSTHG

BA

BAidmix

idmix

idmix

idmix

idmix

+==

−=−=

ωω∆

∆∆∆∆

)xlnxxlnx(RTG BBAAidmix

αααα∆ +=

Equilibrium: phase diagramsThermochemistry of solutions


BABBAA

ABABxsmix

xsmix xx)

2

EEE(NHG λ∆∆ =+−==

REGULAR model of interactive mixing (liquid solution)

2BB )x1(

RTln −= λγ

BB0BB xlnRT γµµ +=

AA0AA xlnRT γµµ +=

Many mixing energetical models exist, which all end with γfunction of various interaction parameters. In all cases:


DLP (δ Lattice Parameter) model (solid solution ex: GaxAl(1-x)As)

2BCB )x1(

RTln −= λγ ( )

5.4

BCAC2BCAC

7

210.5

−

+−= δδδδλ

G.B. Stringfellow J. Crystal Growth 27 (1974) 21.


0=λ

0>>λrepulsionBA0 −>λ

attractionBA0 −<λ



Equilibrium between phases α, β…: they don’t exchange energy, they are at the same energy level.

Constant T and P:

0...)GG(ddG...GG =++=== βαβα

Each chemical component i should be in equilibrium between the various phases

j

ii

n,T,Pii n

G...

∂∂===

ααβα µµµ

0...dndni

iii

ii =++∑∑ ββαα µµ

Equilibrium: phase diagrams


In case of strong repulsion (λ>>0)

Two phases appear (demixion)

A

21

n,T,PBBBB n

G

∂∂==

αααα µµµ





Construction of a 3 solid phase binary phase

diagram

λ<<0 strong attraction: α+β C


Equilibrium: phase diagramsManipulation of phase diagram

AA0AA xlnRT γµµ +=

S.Uda, X. Huang, S. Koh J. of Crystal Growth 281 (2005) 481–491

LangasiteLa3Ga5SiO14

∂∂++= E

2

1

xxlnRT A

AAA

0AA εΩγµµ

500V.cm-1

Congruent growth from the melt!


Equilibrium: point defects


Intrinsic• Empty site : Vacancy , VSi• Misplaced atom : Interstitial (SiI)• Vacancy + interstitial = Frenkel DefectExtrinsic• Foreign atom: Impurity (Insertion or Substitution)

Vacancy

Interstitial

Impurity in Insertion

Substitutional impurity

Frenkel defect




4 broken bonds2 created bonds EV ~ Ecoh /2

Vacancy

4 created bonds + lattice distorsion

EI high

Interstitial

EV = 0.7 to 1 eV (cf. Ecoh)EI = 3 to 5 eV (strong distorsions)




Point defects are thermally activated: thermodynamic defects

.

.

.

.

. .

.

. .

.

N0 atoms, NV vacancies

Ideal solution:

Boltzmann dist.:

→

EV = 1 eVN0 = 6 1023mol-1

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

MixMixVV S T– H E NG ∆∆∆ +=

)N

N( ln kT– E0

dN

Gd

!N!N

)!NNln(kS

0H

V

0V

V

V0

V0Mix

Mix

==

+=

=

∆

∆

∆

kT

E-

0V

V

eN N =




Temperature, K

Vac

ancy

con

cent

ratio

n




µ-void precipitation

E1 = N.EV (-TΔSMix) E2 = σsv4πR2

N < Ncrit = 9/16. π a6 (σsv/EV)3 < N

a8

aVVSi:forNVπR

3

4 3

atomVV3 ===




µ-voids and interstitial precipitation

106 cm-3

InterstitialsVacancies

Annealed under Ar, H2, > 1000°C As in GaAs, very stable

1010 cm-3




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




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




Striations => stoichiometry=> dislocations





EL2 = As in a Ga site

4

V

V

AsEL

AsGaGa0As

Nx

xKxx

e4VAsVAs

As

Ga

Ga2+

−

++ ==

++↔+ −+++−

Important defect: fixes the Semi Insulating character of GaAs

(109cm-3 vs. 1015cm-3 C)

Intrinsic Point defects in a compound: example of GaAsE

CB

VB

EC

Ev

Gap1.43 eV

EL2




As pressure control


Lagowski J., Gatos H.C., Aoyama T., LIN D.G., Appl. Phys. Lett. 45 (1984) 680-682..


3) Out of equilibrium


Driving force and supersaturation

Various Crystal Growth processes


Out of equilibrium


No crystal growth at equilibrium!μμμμ

TTMelt

μLiquid=μCrystal=μ0

Liquid

CrystalΔμ

Δμ = is the driving force for crystallizationΔT is the undercooling

ΔT

Growth rate depends directly onΔμ


Out of equilibrium


( ) ( )

TT

HTS)TT(SdTS

dTT

-dTT

---

melt

MeltMeltmeltMelt

T

T

Melt

T

T

CrystalT

T

Liquid0Crystal0LiquidCrystalLiquid

Melt

MeltMelt

∆∆µ∆∆∆∆∆µ∆

µµµµµµµµµ∆

===−≈=

∂∂

∂∂

=−==

∫

∫∫

Growth from the melt

Growth from vapour

0

P

P

Crystalvapor

P

P

CrystalP

0P

LiquidCrystalVapor

P

PlnkT

)dP-vv(dPP

-dPP

-00

=

=∂

∂∂

∂=== ∫∫∫

µ∆

µµµ∆µµµ∆

Growth from solution0x

xlnkT=µ∆

Undercooling

Supersaturation

Supersaturation


Out of equilibrium


Growth by 2D nucleationCrystal

Mother phase

µ∆σΩπ∆

µ∆Ωσ∆

σπΩµ∆π∆

eGr0

r

G

re2 er-G

2*Nucleus

*Nucleus

2Nucleus

===∂

∂

+=

ΔΔΔΔG

rr*

*NucleusG∆

Example of Si

00 P

Por

T

T ∆∆

P. Rudolph, in: Crystal growth Technology, H.J. Scheel and T. fukuda (eds.) Wiley (2003).

e


4) Application to epitaxy


“Far from equilibrium”?

Driving force controls kinetics

Stoichiometry and composition of layers: physical properties

Layer structure


Application to epitaxy


00 P

Por

T

T ∆∆




Example of GaAs

[ ] [ ]

[ ] [ ]

[ ]4/1

]As[EqGaEq

GaAs

4

333

4PP

aK

GaAsGaAs4

1

GaAsGa)CH(AsH

=

↔+

↔+

[ ] [ ][ ]

[ ]4/1

]As[EqGaEq

4/1]As[Ga

GaAsAsGa

4

4

4 PP

PPlnRT

4

1 =−+= µµµµ∆




ΔμΔμΔμΔμ* kJ.mol-1

0

100

300

200

VPELPE MBEOMCVD

Example of GaAs

“Far from equilibrium” Interface kinetics >> diffusionStringfellow G.B. in “Crystal growth-from Fundamentals to Technology”, Müller G., Métois J.J., Eds. (2004) Elsevier




Generally thermodynamic equilibrium applies at the interface

[ ] [ ]4/1

]As[EqGaEq4/1

]As[Ga 44PPPP >>

The process runs with a strong excess of As: is constant:

[ ] [ ] ]As[GaGaEq 4P4PP <<<<

]As[ 4P

1) The process is controlled by the diffusion of Ga:[ ] [ ]( )

RT

PPDJ GaEqGa

Ga

−=

4) Example of GaxAl(1-x)As: x is totally controlled by [ ]

[ ]Al

Ga

P

P

2) Layer stoichiometry has nothing to do with[ ]

[ ]4As

Ga

P

P

3) Changing the flow rate of [Ga] has no effect on layer stoichiometry


Application to epitaxyManipulation of phase diagram



Application to epitaxylayer structure

Gibbs energy per unit area

G

n

Substrateσ

Bulk deposit

vapor/layer

layer/Substrate

σσ+

Substrate

n

atomic layers

n

Glayern ∂

∂=−µ

n

Gbulk ∂

∂=

>

µ

Stable uniform layer



Gibbs energy per unit areaG

nSubstrateγ

Bulk deposit

vapor/layer

layer/Substrate

σσ+

n

Gbulk ∂

∂=µ

n

Glayern ∂

∂=

>

−µ

Unstable layer

Substrate



Gibbs energy per unit areaG

nSubstrateσ

Bulk deposit

vapor/layer

layer/Substrate

σσ+

layernbulk −> µµ

nc

layernbulk −< µµStable

thin layerfor n<nc

Unstable layer for n>nc

nc

Substrate


Conclusion Transport

How atoms are carried to the interfaceKinetics

How atoms are piled up at the interface

Thermodynamics apply because kinetic>>growth rate


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)?


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

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