Nucleation 1

8/6/2019 Nucleation 1

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Substrate Surfaces &

Thin-Film NucleationBasic modes of thin-film growth


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Thermodynamics ofThin-Film Nucleation

Thermodynamics ofThin-Film Nucleation

Issues addressed:

conditions for stability of thin-film role of surface energies energies involved in nuclei formation thermodynamics of different modes influence of deposition rate & T

Issues addressed: conditions for stability of thin-film role of surface energies energies involved in nuclei formation thermodynamics of different modes influence of deposition rate & T


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Thermodynamics of Thin-Film Nucleation

r

sv

fv

fs

nucleusdeposition

deso

rp

t

ion

substrate

cosffsv vs +=

)(sin)cos-(12

)coscos32(3

)(

svfs

22

fv

2

V

33

++

++=

rr

GrrG

V

SfsSfv

~ 0.2 3 J/m2ignore


4/15


5/15


o90;fsfvsv >+<

metals on dielectrics

fv

fssvcos

=

no wetting

wetting

( ) o90;fsfvsv

autoepitaxy: fs=0

In general, materials with low surface energy

will wet substrates with a higher surface energy

misfit


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eR

RTkG

&

&

lnBV =

equilib. evap. rate from the nucleus at

the substr.T

K3

6

V

,22310

V

svfsfsfvfv

J/m108/

mJ/m05.0/fv,J/m1fv,J/m106.1Vfor

03

))((2*

=


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Zoo of adatomsfate

Kinetics of Thin-Film NucleationIssues addressed: nucleation rate and its dependence on

t, T, , film itself and substrate

development of film growth and coalescence of nuclei

t-dep. of the growth and coalescence of nuclei

binding (good): physisorption:

van der Waals (weak, 0.01eV)

chemisorption:chemical bonds (strong, 1-10eV)

defects of

substrate

reflection (bad)

homogeneous

nucleation

heterogeneous

nucleation

( referred)

desorption (bad)

hopping & diffusion(initial growth)


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Tk

EaD

B

diff2

0S exp~

Kinetics of Thin-Film Nucleationdiffusion is determined by: Ediffenergy barrier for diffusion

Ts temperature of substrate vibrational frequency of an adatom

(attempt frequency)

=

Tk

E

B

diffJ expfrequency of surface jumps

X

cluster can be formed in overlap region

surface diffusion coefficient

=

Tk

EEaDX

B

diffdes0SS

2

exp~

~a0


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Kinetics of Thin-Film Nucleation

desorption is determined by: Edes energy barrier for desorption Ts temperature of substrate

vibrational frequency of an adatom

(attempt frequency)

= TkE

B

des

S exp

1

lifetime of desorbing adatoms


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Kinetics of Thin-Film NucleationNucleation rate

)s(cm**2-

ANN =&

critical area

impingement rate onto nucleusequilibrium concentration

ofstable nuclei

==

RT

Mv

RT

M

v

n

nvf x

x

xx

2exp

2d

d1)(

2

Maxwell-Boltzmann

RMT

PN

M

RTnv

RT

Mvv

RT

Mnvvfnnv

xx

xxxxx22

d2

exp2

d)(d A2

000

==

===

=

Tk

GnN

B

S

*exp*

RTN

nP

A

=

the perfect gas law

~X

r*sin

a0( ) 0sin*2* arA

=

Tk

E

B

diffJ exp

JS =

(absorption flux from vapor)

=

Tk

E

B

desS exp

1

( )

=

Tk

GEEn

RMT

PNarN

B

diffdesA0

*exp

2sin*2 &


11/15

Kinetics of Thin-Film NucleationAtomistic Nucleation Model

+=

*

*1B

** lni

i

iiN

NTkEGCritical free energy to form nucleus ("molecule")

(chemical equilibrium between clusters and monomers)

~X

r*sin

a0

=

Tk

E

B

diffJ exp

=

Tk

E

B

desS exp

1

Walton-Rhodin Theory

treats clusters of atoms as molecules rather than solid caps

considers the bonds between atoms is similar to the capillarity model

Ei* - energy to break/form a critical cluster of i* atoms

Ni* - density of critical nuclei

N1 density of single adatoms

n0 density of adsorption cites"Chemical" reaction: iA=Ai

==

Tk

ERRN

B

desS1

exp

&

&

=

Tk

EEaDX

B

diffde s0SS

2exp~

=

Tk

EEaXR

B

diffde s2

0

2 exp&

++

=

Tk

EEEi

n

RnaRN i

i

i

B

*diffdes

**

0

0

2

0*

)1(exp

&&&


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Law of Mass Action (Waage & Guldberg 1867)

aA + bB + ... xX + yY + ...

[forward rate] k 1 [A]a [B]b ...

[reverse rate] k -1 [X]x [Y]y ...

in equilibrium, k 1 [A]a [B]b ... = k-1 [X]

x [Y]y ...

H2 + I2 2HI;the direct reaction results from collision of H2 and I2 molecules =>

reaction rate is proportional to the number of such collisions;

the number of collisions is proportional to density of H2 and I2;

the density is proportional to pressure =>

the reaction rate is proportional to the partial pressures of H2 and I2 :

k1 PH2 PI2 similarly, the reverse reaction rate is proportional to the number of collisions

between HI molecules => the reaction rate is

k-1 PHI2

in equilibrium k1 PH2 PI2 = k-1 PHI2

we define the constant of equilibrium as

K(T) = k-1/ k1 = PH2 PI2/PHI2

G = G0 + RTlnK,


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Kinetics of Thin-Film NucleationAtomistic Nucleation Model

++

=

Tk

EEEi

n

RnaRN ii

i

B

*diffdes*

*

0

0

2

0*)1(exp

&&&

(continued)

+

=

n

Rk

EET &

lnB

2des21

2*1* == = ii NN &&

+=

21B

2des0 exp

Tk

EEnR &


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Kinetics of Thin-Film NucleationKinetic Nucleation Model

=

=2

111

S

11

d

d

i

iiNKNNKN

RNt

&

iiiii NNKNNKNt

1111dd =

=

Tk

E

n

RAnN

p

B0

0S exp

&


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Kinetics of Thin-Film NucleationCluster Coalescence and Depletion

r1

21

r2

Mass transport

iii

ii

i

i

i

i

rdrr

drrrd

rd

dn

dG

=

=

=

==

2

)34(

)8()34(

)4(

2

3

2

Ostwald rippening (large islands grow atthe expence of the smaller ones)

Sintering Cluster migration

Grain size

Nucleation 1

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