薄膜力学-3-生长模式

γ = γ 0 + fαβ ε αβ
S
Reference state: unstrained surface!
A liquid membrane experiment
L
Variation of free energy:
W F
δG = 2γWδL − FδL
No friction
By principles of thermodynamics: In equilibrium:
t
Cluster coalescence
• Ostwald ripening: the desire to minimize surface energy drives atoms to diffuse from smaller islands to larger islands; larger islands grow or “ripen” at the expense of smaller ones. • Sintering: a neck forms when two clusters are in contact; atoms diffuse to the neck to reduce the total surface energy. • Cluster migration: clusters migrate on substrate surface and coalesce when they collide.
N
B C A
A: nucleus density increases B: nucleus density saturates at a stable value, depending on deposition rate and substrate temperature C: nucleus density decreases due to coalescence
∆G*
Homogeneous nucleation (θ = 180°)
θ = 0, ∆G* = 0
θ
Kinetics of nucleation
• Nucleation rate:
dN = N * A*ω dt
N*: equilibrium concentration of stable nuclei (per unit area) A*: critical area for atomic impinging ω: rate of atomic impinging (per unit area per unit time)
EM 397: Thin Film Mechanics
III. Growth Modes of Thin Films
Nanshu Lu
The University of Texas at Austin Fall 2013
Growth modes
• Layer-by-layer mode (Frank-van der Merwe or FM)
δG = 0
F = 2γW
Otherwise, free energy tend to decrease: δG < 0
F > 2γW F < 2γW
How about an elastic membrane?
δL > 0 δL < 0
Interface energy
• Same physical origin as surface energy • Consider a spherical cap cluster on a substrate:
– Initially FM growth – After a few monolayers, other energy contributions (e.g., strain energy) trigger the transition from layer to island growth.
γ s ≥ γ f + γ sf γ s < γ f + γ sf
Exposed bulk plane (unbalanced surface)
Surface atoms relaxed outward
Reconstruction of surface layers
Surface energy
• Atoms at a free surface are more energetic than atoms within the bulk • Create two surfaces by cutting a large block into two parts:
– free energy reduces by condensation (per atom):
∆GV = −kT ln p = −kT ln(1 + S ) pe
k = 1.38 × 10-23 J/K = 8.617 × 10-5 eV/K
Thermodynamics of nucleation
γf θ
2R
γsf
γs
Free energy:
G ( R, θ ) = γ f Af + (γ sf − γ s )πR 2
γf γs γsf
2πR 2 Af = 1 + cos θ πR 3 2 + cos θ 1 − cos θ Vf = 3 1 + cos θ sin θ
δG = 0 δVf = 0
∆G =
γf θ
2R
Vf
Ω
∆GV + γ f Af + (γ sf − γ s )πR 2
∆GV = −kT ln p pe
γsf
γs
πR 3 2 + cosθ 1 − cosθ Vf = 3 1 + cosθ sin θ
2πR 2 Af = 1 + cosθ
Critical nucleus
∆G ∆G* 0
– Homoepitaxy, e.g., Si on Si
• Island mode (Volmer-Weber or VW)
– Metal or semiconductor on oxide substrates
• Layer-plus-island (Stranski-Krastanov or SK)
γ f + γ sf > γ s > γ sf
γ s ≥ γ f + γ sf
θ
γ s < γ sf < γ s + γ f
γ sf ≥ γ s + γ f
Film growth modes
• In macroscopic terms, the three film growth modes can be understood based on the surface energies or simply the wetting angle. • FM mode: • VW mode: • SK mode:
– Heteroepitaxy, e.g., Ge on Si, GaN on saps of solid surfaces
• Substrate surface plays a very important role in the film growth process and influences the structures and properties of thin films. • Metal: increased electron density near a surface • Semiconductors: dangling bonds at a surface • Surface relaxation and reconstruction
∂∆G =0 ∂R
R* = 2γ f Ω − ∆GV sin θ
and
R R*
2 Ω 16πγ 3 (2 + cosθ )(1 − cosθ ) 2 f ∆G* = 3(∆GV ) 2 4
0 <θ <π
To reduce the free energy, nucleus shrinks and disappear if R < R*, and grows if R > R*. ∆G* is the barrier energy for nucleation. Equilibrium concentration of stable nuclei is
θ =0 θ >0
Vapor condensation
• At equilibrium pressure, atoms attach to and detach from the solid surface with equal probability, thus no net deposition. • When the vapor pressure is higher (supersaturated), more atoms attach to the solid surface
2 16πγ 3 Ω f
3(∆GV ) 2
• Heterogeneous nucleation:
– Substrate catalyze nucleation by lowering the energy barrier through the wetting angle – Strain energy due to lattice mismatch increases the barrier – Surface defects (crystalline steps, grain boundary traces, dislocations, impurities, etc.) could decrease the barrier.