Chernov ISSCG 14

8/13/2019 Chernov ISSCG 14

1/42

[email protected]

Lawrence Livermore National Laboratory, P. O. Box 808, Livermore, CA 94551

Thiswork

performed

under

the

auspices

of

the

U.S.

Department

of

Energy

by

LawrenceLivermoreNationalLaboratoryunderContractDEAC5207NA27344

Crystal Growth andCrystal Growing

Santander Spain, July 20-24, 2009

Lawrence Livermore National Laboratory

SurfacePhenomena

and

Parameters

ofCrystalGrowth:SimpleBasics

ISSCG14, DailanChina,August172010

LLNL-PRES-445231


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Outline Interface:SmoothversusRoughState

Nucleation

GrowthKinetics:

SmoothInterface

RoughInterface

Growthmodes

and

as

grown

defects


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Bibliography

Monographs A.A.Chernov. CrystallizationProcesses. ModernCrystallographyIII.

SpringerSer.SolidState,vol36,1984(basicmolecularandmacroscopicphenomena)

F.Rozenberger.FundamentalsofCrystalGrowth. Springer,1976(phase

diagrams,transport,

convection)

A.Pimpinelli,J.Villain. PhysicsofCrystalGrowth.CambridgeUnivPress,1998(generalphysicalapproach)

K.A.Jackson.KineticProcesses.CrystalGrowth,Diffusion,andPhaseTransitioninMaterials. WileyVCH,Weinheim,2004

J.W.Mullin.Crystallization.

Butterworths,

2001

(4

dedition)

(growth

from

solution,industrialcrystallization)

D.Kashciev. Nucleation.BasicTheoryWithApplications.Butterworth/Heinemann,2000

I.Gutzov,J.Schmelzer.TheVitreousState. Thermodynamics,Structure,Rheology,

and

Crystallization.

Springer,

1995

I.Markov. CrystalGrowthforBeginners.WorldScientific,1996

J.A.Venables.SurfaceandThinFilmProcesses.CambridgeUnivPress,2000


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J.N.Israelachvili.Intermolecularandsurfaceforces.Academicpress,1992

A.Zangwill.PhysicsatSurfaces. CambridgeUnivPress.1988

MonograpicReviewes

CurrentTopicsidMaterialsScience. Ed.E.Kaldis.NorthHolland,Vols110,1976

1982.

Crystals.Growth.Properties.Applications.Ed.H.C.Freihardt.Springer,Vols113,19781991.

HandbookofCrystalGrowth.EdD.T.J.Hurle,vols1A,B,2A,B,3A,B.NorthHolland,199394

SolidsFar

From

Equilibrium.

Ed

C.Godreche.

Cambridge

Univ

press,

1992

BulkCrystalGrowthofElectronic,OpticalandOptoelectronicMaterials.Ed.P.Capper.J.Wiley&Sons,2005


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InternationalSummerSchoolsonCrystalGrowth

Crystal Growth AnIntroduction. EdP.Hartman.NorthHolland

1973

CrystalgrowthandCharacterization. Ed.R.Ueda,J.B.Mullin.North

Holland.1974

CrystalgrowthofElectronicMaterials.Ed. E.Kaldis.NorthHolland,

1985

ScienceandTechnologyofCrystalGrowth. Ed.J.P.vanderEerden,

O.S.L.Bruinsma.

Kluwer

1995TheoreticalandTechnologicalAspectsofCrystalGrowth. Ed.

R.Fornari,C.Paorichi. TransTechPublications1998

AdvancesinCrystalGrowthResearch. Ed.K.Sato,Y.Furukawa,

K.Nakajima. Elsevier,2001

CrystalGrowth

From

Fundamentals

to

Technology.

Ed.

G.Muller,

J.J.Metois,P.Rudolf. Elsevier2004


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Smooth and Rough Interface

cubicle approximation

H.J.Leamy,G.Gilmer, K.A.Jackson. In: Surfaces of Materials vol 1 p.121 (1975)

/2kT


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Non-Kossel crystalwi

i = 3

31

2

w

w+

b

c

w-

Kossel CrystalKink

positions of

a molecule

InterfaceGeometry. KinkPosition.

Smooth interface


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S.Toshev,inCrystalgrowthAnIntoduction,EdP.Hartman,NH,1973p.1

m=NA2/3 molar

surface

free

energy

cal/gatom

Enthalpyofmelting,H, cal/gatom

Slope=0.30.5

Fromhomogeneousnucleationinmelts(D.Turnbull):

Interfaceenergy: SL /22/3 h/62/30.17h/2/3

LVLSLVSLSV 3/23/2 )/(13.1)/(13.1

Empirical rule for the crystal melt surface energy


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SurfaceReconstruction,

Si

(001)

Surface energy, Si, crystal vacuum: /22/3 2,600 erg/cm2, saturation oftwo bonds releases 1,000 erg/cm2 (40%) , thus 1,600 erg/cm2.

Experiment, computations: 1,490 1,623 erg/cm2.

M.Lagally et alTypically, no reconstruction in condensed surrounding or in adsorbing gases.


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Chemically Complex Solutions: both h < 0 and h > 0.

Melting and evaporation always require heat supply.

Dissolution:Na2SO4 in water generatesh = 0.28kcal/mol (= 1.9.10

-14erg per molecule)

Na2SO4.10H2O requires h = 18.7kcal/mol.

Mn(NO3)2 generatesh = 12.7kcal/mol,

Mn(NO3)2.7H2O requires h = 6.1kcal/mol.

MgI2, K2CO3, CaBr2 generates 50.2, 6.9, 26.3kcal/mol

The reason hydration (solvation) of ions.

Hydration enthalpy, (ion-dipole electrostatic attraction) x (N of H2O molecules)

Experiment: Na+ 101 kcal/mol, Ca2+ 386 kcal/mol,

SO42- 265 kcal/mol, NO3 74 kcal/mol

Nafor1155.9.225.).(

2 mol

kcal

mol

kcal

r

elZe


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TTm

uL

uS

sLsS

Melt

Solid

T

uL

uS

Liquid,C2>C1

C1 C2

T2T1

Liquid,C2>C1

C1C2

uL1

uL2

uS

TT1T2

Solutions retrograde solubility

Regular melting

Regular solubility Retrograde solubility


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/2kT -0.272 lnCe (mol/m3) + 2.82.


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Step meandering fluctuations by attachments -detachments

2


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( )

Roughening transition: Gstep = UstepTSstep = 0

kTw

e


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Roughening transition viewed by Monte Carlo simulation

meandering rises as /2kT decreases

H.J.Leamy,G.Gilmer, K.A.Jackson in Surfaces of Materials vol 1 p.121 (1975)

/2kT =


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4He Crystal in its melt

1.4K

1K

0.4K

0.1K

S.Balibar, H.Alles, A.Ya. Parshin.

Rev Mod. Phys. 77(2005)317-370


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KH2PO4, KDP CaCO3, Calcite Lysozyme

Step fluctuation viewed by AFM: meanderingincreases together with solubility and decreasing step energy

J.J.DeYoreo, L.N.Rashkovich, R Friddle


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Nucleation:

3Dimensional,

2Dimensional


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Critical

radius:

G

/r

=

0:

rc=

2/Gc=(16/3)

23/2 3Dimensions,sphere

Nucleationis

solids

elastic

stress:

G=(4r3/3)[9(1)(P)2/16E]+4r2Ifelasticstressissignificanttheequilibriumshapeisaflatdisc

orablade

Nucleation rate: J (1/cm3s) = Bexp(- Gc/kT)Nucleation work is minimal for equilibrium shape. Sphere:

G = - (4r3/3) + 4r2

r

srf

3D Homogeneous Nucleation

M -S = kTlnC/Ce - solution = ST = H(T/T) - melt

G

rc=2/

Gc

Classical

Quantum


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Pre-exponent, B

Homogeneous nucleation:B = 4rc2an2exp(E/kT)Z= 4rc2n2Z,

Zeldovich factor: = - (1/nc)(2G(n)/n2)nc,

Z = (/2kT)1/2 = (1/nc)(Gc/3kT)1/2~ 10-2

Heterogeneous nucleation .

Equilibrium shape on a substrate is a part of the free equilibrium shapebecause of the Gibbs-Thomson law:

Estimated B:

vapor B ~ 10281/cm3s; melt B ~1035 1038 1/cm3s, solution B ~ 1027 1/cm3s

Ghet = Gc/

h

L

h/L = /2

hc = 2/

+ s -is

i


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Non steady state nucleation - time lag

G = - n + bn2/3

nc = (32/3)23/()3

n

Gc

Number of nuclei

time

0

n

0 1 2 3 .

N = (Nucleation rate J).Volume.t

Diffusion along the n-axis:

D* = +

+ -

Time to reach maximum of the

nucleation barrier:

~ nc2/D*

(Nucleation Rate J).Volume


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G

rc= /

r

Gc

2D Nucleation

G = - (r2h/) + 2rh

rc= /

Gc = h2

/

New atomic layer, one atomic size or lattice

spacing high, h:

Homogeneous 2D nucleation rate:

J(1/cm2s) = 2rcanexp(-E/kT)nZexp(-Gc/kT) = 2rcn2exp(-Gc/kT)

Heterogeneous 2D nucleation rate: n2 nn*, Gc Gc*

rc+a

rc


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Growth Kinetics

Smooth Interface Layer-byLayer GrowthGeneration of steps:

3D nucleationScrew dislocation


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Nucleation rate, J (nuclei/cm2s = 1/cm2s)

Step expansion rate, v cm/s

Low nucleation rate, small face of the area S: R = JSh

High nucleation rate, large area S: R = h(v2J/3)1/3

h

R rate of the face propagation

SActivenucleation site

v

2 Dimensional nucleation


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M = 14,000 Da, effective

molecular diameter ~ 2.5nm

2D nucleation on lysozyme


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Step Sources: Screw Dislocations

D

R = pv=hv/

h/=p

Face growth rate R:


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(0001) SiC

St P ti Ferritin M = 450 000 Da spheres 13nm


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P.G.Vekilov et al

Step Propagation High Kink Density.

Ferritin, M = 450,000 Da, spheres 13nm,

FCC lattice.

Supply of species to steps and incorporation

into kinks.

1. Surface/at-surface diffusion: vapor, MBE,CVD, etc

2. Bulk diffusion: solution

steps

kinks

(111)


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Incorporation into a kink

,iu

kTi

p Ae

1, sum over medium or solid, except top of the barrier,iu

kTi

i

Ae u E

,ln)(lnlnlnT

uAk

kT

uAAekppkpks ikT

u

iii

i

.,ln kT

Tsu

eAuAkTTs

kT

E

kT

u

ikT

uTsu

i eepepii

pressure,constantator,,

:

Activated complex includes:

The still hydrated molecule/ion to be incorporated, the hydrated kink and their

surrounding,

or

The molecule/atom/ion to be incorporated still bound within the species that havecarried it to the kink and the kink occupied by other species, if any.

Probability of having the activated complex at the energy level ui :


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Kinetic coefficient

kT

Eu

k

ekkTkT

E

kT

E

kT

E

M

SMSSM

ea

CCeeaeeawwa

),()1()()(vk

MeS uCkT ln

aCw

C

CCe

w

wek

e

ekT /fluxexchangetheand

Attachment and detachment frequencies of species to a kink:

kT

E

kT

E SM

ewew

,

kT

uE

kstestst

M

eaCC

2 )/(),(v

Kink velocity:

Step velocity:

Face velocity: )(vst est CCppR


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J.Q.Broughton,A.Bonisent,F.F.Abraham.J.ChemPhys74(1981)4029

Delocalizedinterface


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J.Q.Broughton,A.Bonisnt,F.F.Abraham.J.ChemPhys74(1981)4029


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Meltnearsoidwall densitywaves

Several atomic planes are


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Severalatomicplanesare

beingpacked

simultaneouslyMikheev,Chernov,JCG112(1991)591

Normalgrowthrate:

V=bT

=A(kT/m)

1/2

(T/T),

A1; b=A(k/mT)1/2

Lead:

m=207.1.67.1024g,

T=600K,

(kT/m)1/2 =155m/s

b=26

cm/Sk

Experiment:

b=288cm/sK

Rodway,Hunt

JCG

112(1991)554


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Growth modes andas grown defects


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See K.F.Hulme, J.B.Mullin. Phil Mag 22(1959)1286

K = 0.5

K = 4


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Sectoriality and Striation

Vicinal Sectoriality


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Vicinal Sectoriality


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Bottom Lines

Crystal interface may be either smooth or rough

at high or low energy, respectively.

Surface energy may be estimated from phase transformation

enthalpy for vapor and melt and from solubility for solution.

Growth of a smooth interface is impossible without kinks atsteps and is thus slower than growth of the rough interface full

of kinks or is fully disordered

Growth rate of a smooth interface is controlled by stepgeneration by 2D nucleation or by screw dislocations and by

incorporation of growth units at kinks.

Growth mode controls crystal perfection

Documents

Chernov ISSCG 14