# Nucleation Concept of Surface Energy

## Transcript Of Nucleation Concept of Surface Energy

Nucleation – Concept of Surface Energy

Differential Energy dE cost for creating a surface are dA

dE = TdS − PdV + γdA dG = −SdT +VdP + γdA

dG = γdA

dE = δW = γdA

Fdx = γdA = γ ( 2b )dx

γ=F 2b

Force per unit length of the circumference (N/m) is the surface tension and in case of the liquids surface energy per unit area (J/m2) is the same as the surface tension

Physical understanding of the origin of surface energy

! Why is there a surface energy associated with solid surfaces? ! What is the physical explanation/interpretation? ! No bonds from the top. Compared to bulk the bonding may not be

satisfied. ! Surface energy tries to minimize itself by diffusion. If surface

diffusion is fast the nuclei will expose the lowest energy surfaces. ! γ depends on

⇒ Chemical composition

⇒ Atomic scale roughness

⇒ Atomic scale reconstruction

⇒ Crystallographic orientation ! Example: (111) plane of an FCC structure is the closest pack plane.

The # of bonds in the plane is maximized and the number out of the

plane are minimized. Consequently

γ

111 fcc

<

γ

100 fcc

1

Physical understanding of the origin of surface energy

! Charge neutrality @ the surface leads to lower γ. ! In crystals having ionic bonds (NaCl, CaF2) or polar bonds (GaAs,

ZnO) g tends to be lower for faces that contain equal numbers of

cations (Na, Ca, Ga, Zn) and anions (Cl, F, As, O). ! i.e., nonpolar surfaces are lower energy. ! γ for a particular surface may be lower than predicted from the

dangling bond density because crystal surfaces rearrange (called

reconstruction) such that termination does not necessarily have the

geometry of the atomic arrangements of the cleaved plane in the

bulk crystal.

! γ is lower for low index planes called singular surfaces

θ

γ

θ

Nucleation

Phase 3 Phase 2

Phase 1

Phase 3

Phase 2

or

Phase 1

At constant P and T dG < 0 for nucleation to occur

Phase 2 Phase 1

create area dA of 1-2 and 2-3 interface

Phase 3

Gbefore − Gafter = [γ 13 − ( γ 12 + γ 23 )]dA

[γ 13 − ( γ 12 + γ 23 )]≥ 0 [γ 13 − ( γ 12 + γ 23 )]< 0

Phase 2 spreads over phase 1 completely “wetting” it

Phase 2 remains in a lens configuration

2

Film Growth Modes

γ 13 = γ s γ 12 = γ i γ 23 = γ f

γ s ≥ ( γ + γ f ) Frank van der Merwe

i

(layer by layer growth)

γs <(γ i +γ f )

Volmer-Weber (island growth)

γ s ≥ ( γ i + γ fo ) γs <(γ i +γ f )

1st later

StranskiKrastanov growth

Nucleation: the critical radius

γs γ f r

Assume that γ s < ( γ i + γ f )

∆G

=

(

µ

s

−

µ

v

)

1 ~

4 πr3 + γ f 4πr 2

Vs 3

∆G = −∆µ

1 ~

4 πr3 + γ

4πr 2

where

∆µ = µ v − µ s

V3

f

s

∆G

∆G*

Growth after rc decreases the ∆G Such that nuclei that have r>rc grow Nuclei with r

rc

r r = 2γ f ~

c ( ∆µ / Vs )

3

Nucleation rate

P

L S

P Psat

V

Tsat

T

∆G* = 16

πγ

3 f

~

3 ( ∆µ / Vs )2

∆µ = µ v − µ s µ v = µ o( Po ,T ) + RT ln( P / Po ) at P = Psat µ s = µ v = µ o( Po ,T ) + RT ln( Psat / Po )

∆µ = RT ln( P / Psat )

∆G* = 16

πγ

3 f

3 RT ln( P / Psat )2

V~ s

Another way of looking at it

∆µ ∝ ∆T

rc ∝ 2γ f ∆T

∆G* ∝

γ

3 f

( ∆T )2

Nucleation rate

Rn ∝ e−∆G* / RT

Nucleation morphology

γf

γs θ

γi

substrate

Assuming amorphous or liquid nuclei one can do a force balance in the direction parallel to the surface

γ s = γ i + γ f cosθ

While we can equate surface energy with surface tension in liquids this is not the case with solids. For solids there is also a quantity called surface stress.

In solids surface energy is anisotropic and the shape of the nuclei will adjust to minimize the energy of the total surface area through

∑γ k Ak = min imum k

4

Nucleation morphology

(010) (110) ro

r1 (100)

Consider the case of poor wetting Ai=0

substrate

∑γ k Ak = min imum k

“Wulff Theorem”

γ k = constan t rk

! γ1 = 1.2 γo ! Higher energy surfaces tend to grow

faster. They tend to be higher index surfaces. ! Faster growing faces grow themselves out of existence

Island Nucleation & 2-D nucleation

γ 11

γ 11

γi

γ 01 << γ 11 ≈ γ i

γ 01 γ 10 γ 10

γ 10 << γ 01 ≈ γ i γ 10 << γ 01 < γ i

γ 01

γ 10

γ 10

γi

γ 01 ≈ γ 10 ≈ γ i

2-D Nucleation

Frank van der Merwe (layer by layer growth)

5

Epitaxy

Film crystallinity

Ea~5 eV 6

Film Morphology - Growth Zones

! Z1: Ts/Tm so low that surface diffusion is negligible (Λ ! ZT: like Z1 but domes and voids are absent. Common in energy-enhanced deposition methods (plasma, sputtering, etc.)

! Z2: occurs for Ts/Tm>0.3, surface diffusion is significant, Tight columns with grain boundaries, column diameter ↑with Ts ↑. Columns less defected than Z1 or ZT and faceted at top. The boundaries are planes of reduced bonding rather than planes of xtallographic discontinuity.

! Z3: occurs for Ts/Tm>0.5, significant bulk annealing takes place, isotropic and equiaxed xtals. Films are smoother.

7

Differential Energy dE cost for creating a surface are dA

dE = TdS − PdV + γdA dG = −SdT +VdP + γdA

dG = γdA

dE = δW = γdA

Fdx = γdA = γ ( 2b )dx

γ=F 2b

Force per unit length of the circumference (N/m) is the surface tension and in case of the liquids surface energy per unit area (J/m2) is the same as the surface tension

Physical understanding of the origin of surface energy

! Why is there a surface energy associated with solid surfaces? ! What is the physical explanation/interpretation? ! No bonds from the top. Compared to bulk the bonding may not be

satisfied. ! Surface energy tries to minimize itself by diffusion. If surface

diffusion is fast the nuclei will expose the lowest energy surfaces. ! γ depends on

⇒ Chemical composition

⇒ Atomic scale roughness

⇒ Atomic scale reconstruction

⇒ Crystallographic orientation ! Example: (111) plane of an FCC structure is the closest pack plane.

The # of bonds in the plane is maximized and the number out of the

plane are minimized. Consequently

γ

111 fcc

<

γ

100 fcc

1

Physical understanding of the origin of surface energy

! Charge neutrality @ the surface leads to lower γ. ! In crystals having ionic bonds (NaCl, CaF2) or polar bonds (GaAs,

ZnO) g tends to be lower for faces that contain equal numbers of

cations (Na, Ca, Ga, Zn) and anions (Cl, F, As, O). ! i.e., nonpolar surfaces are lower energy. ! γ for a particular surface may be lower than predicted from the

dangling bond density because crystal surfaces rearrange (called

reconstruction) such that termination does not necessarily have the

geometry of the atomic arrangements of the cleaved plane in the

bulk crystal.

! γ is lower for low index planes called singular surfaces

θ

γ

θ

Nucleation

Phase 3 Phase 2

Phase 1

Phase 3

Phase 2

or

Phase 1

At constant P and T dG < 0 for nucleation to occur

Phase 2 Phase 1

create area dA of 1-2 and 2-3 interface

Phase 3

Gbefore − Gafter = [γ 13 − ( γ 12 + γ 23 )]dA

[γ 13 − ( γ 12 + γ 23 )]≥ 0 [γ 13 − ( γ 12 + γ 23 )]< 0

Phase 2 spreads over phase 1 completely “wetting” it

Phase 2 remains in a lens configuration

2

Film Growth Modes

γ 13 = γ s γ 12 = γ i γ 23 = γ f

γ s ≥ ( γ + γ f ) Frank van der Merwe

i

(layer by layer growth)

γs <(γ i +γ f )

Volmer-Weber (island growth)

γ s ≥ ( γ i + γ fo ) γs <(γ i +γ f )

1st later

StranskiKrastanov growth

Nucleation: the critical radius

γs γ f r

Assume that γ s < ( γ i + γ f )

∆G

=

(

µ

s

−

µ

v

)

1 ~

4 πr3 + γ f 4πr 2

Vs 3

∆G = −∆µ

1 ~

4 πr3 + γ

4πr 2

where

∆µ = µ v − µ s

V3

f

s

∆G

∆G*

Growth after rc decreases the ∆G Such that nuclei that have r>rc grow Nuclei with r

rc

r r = 2γ f ~

c ( ∆µ / Vs )

3

Nucleation rate

P

L S

P Psat

V

Tsat

T

∆G* = 16

πγ

3 f

~

3 ( ∆µ / Vs )2

∆µ = µ v − µ s µ v = µ o( Po ,T ) + RT ln( P / Po ) at P = Psat µ s = µ v = µ o( Po ,T ) + RT ln( Psat / Po )

∆µ = RT ln( P / Psat )

∆G* = 16

πγ

3 f

3 RT ln( P / Psat )2

V~ s

Another way of looking at it

∆µ ∝ ∆T

rc ∝ 2γ f ∆T

∆G* ∝

γ

3 f

( ∆T )2

Nucleation rate

Rn ∝ e−∆G* / RT

Nucleation morphology

γf

γs θ

γi

substrate

Assuming amorphous or liquid nuclei one can do a force balance in the direction parallel to the surface

γ s = γ i + γ f cosθ

While we can equate surface energy with surface tension in liquids this is not the case with solids. For solids there is also a quantity called surface stress.

In solids surface energy is anisotropic and the shape of the nuclei will adjust to minimize the energy of the total surface area through

∑γ k Ak = min imum k

4

Nucleation morphology

(010) (110) ro

r1 (100)

Consider the case of poor wetting Ai=0

substrate

∑γ k Ak = min imum k

“Wulff Theorem”

γ k = constan t rk

! γ1 = 1.2 γo ! Higher energy surfaces tend to grow

faster. They tend to be higher index surfaces. ! Faster growing faces grow themselves out of existence

Island Nucleation & 2-D nucleation

γ 11

γ 11

γi

γ 01 << γ 11 ≈ γ i

γ 01 γ 10 γ 10

γ 10 << γ 01 ≈ γ i γ 10 << γ 01 < γ i

γ 01

γ 10

γ 10

γi

γ 01 ≈ γ 10 ≈ γ i

2-D Nucleation

Frank van der Merwe (layer by layer growth)

5

Epitaxy

Film crystallinity

Ea~5 eV 6

Film Morphology - Growth Zones

! Z1: Ts/Tm so low that surface diffusion is negligible (Λ ! ZT: like Z1 but domes and voids are absent. Common in energy-enhanced deposition methods (plasma, sputtering, etc.)

! Z2: occurs for Ts/Tm>0.3, surface diffusion is significant, Tight columns with grain boundaries, column diameter ↑with Ts ↑. Columns less defected than Z1 or ZT and faceted at top. The boundaries are planes of reduced bonding rather than planes of xtallographic discontinuity.

! Z3: occurs for Ts/Tm>0.5, significant bulk annealing takes place, isotropic and equiaxed xtals. Films are smoother.

7