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We consider a continuum model for the evolution of an epitaxially-strained dislocation-free anisotropic thin solid film on isotropic deformable substrate in the absence of vapor deposition. By using a thin film approximation we derived a nonlinear evolution equation. We examined the nonlinear evolution equation and found that there is a critical film thickness below which every film thickness is stable and a critical wave number above which every film thickness is stable.

Spontaneously formed periodic domain structures of nanoscale islands (quantum dots) in epitaxially strained thin solid films have become a subject of intense theoretical and experimental study. These islands have unique, optical, electronic and magnetic properties which signify their importance in quantum dot applications [

In experimental study of the nonlinear evolution of the stress-driven instability of thick films the formation of deep, cusp-like grooves was observed [

In a thin film, however, cusp formation is suppressed as the surface approaches the film substrate interface. The different stress fields and the different surface energies of the film and the substrate affect the surface morphology and the film-substrate interface is prevented from being exposed when the wetting criterion is satisfied. Stranski-Krastanow islands will be formed in this case [

While the understanding of some of the theoretical and modeling issues is well developed, the implementation of the models as large-scale numerical simulations is not yet feasible. The central issue is that the dynamics of the surface morphology is coupled to the elastic strain in the system, so dynamic models require solving the elasticity problems throughout the film and substrate at each time step and are limited by storage limitations for 3-dimensional problem. Simulations involving the full elasticity problem are limited to one or few islands [

In [

The rest of the paper is organized as follows. In Section 2 we present the full nonlinear model for morphological evolution in thin solid films. In Sections 3 and 4 we describe the thin-film scalings and derive a systematic approximation to find the dominant terms in thin film evolution. In Section 5 we analyze the stability properties of this reduced equation and finally in Section 6 we summarize our results.

We begin with the model from [

σ i j = c i j k l E i j (1)

where c i j k l is the array of elastic stiffness constants. Each of the nine equations for a stress component involves nine material parameters. The fourth-order tensor c i j k l comprises 81 components. The symmetry of the stress and strain tensors further imply that the components of the stiffness tensor must satisfy c i j k l = c i j l k = c j i k l . As a consequence, the number of independent elastic constants is reduced from 81 to 36. Using this fact and rewriting the stress and strain using their symmetric properties, 1 can be written in a simpler format

σ i = c i j E i . (2)

Cubic symmetry is a property of crystals that possess three fourfold axes of rotational symmetry, the cube axes, and four threefold axes of rotational symmetry, the cube diagonals. Alternatively, cubic symmetry may be described as invariance of material structure under a translation of a certain distance in any of three mutually orthogonal directions; these directions are usually identified as the cube axes. Consider a cubic material for which the [

c 11 = c 22 = c 33 ; c 12 = c 23 = c 31 ; c 44 = c 55 = c 66 (3)

All the other elastic constants vanish because of the fourfold rotational symmetry of the reference axes. Hence elastic response of any cubic crystal is characterized by three independent elastic constants and the stress-strain relationship is given by

σ 11 = c 11 E 11 + c 12 E 22 + c 12 E 33 (4)

σ 22 = c 12 E 11 + c 11 E 22 + c 12 E 33 (5)

σ 33 = c 12 E 11 + c 12 E 22 + c 11 E 33 (6)

σ 23 = 2 c 44 E 23 (7)

σ 31 = 2 c 44 E 31 (8)

σ 12 = 2 c 44 E 12 (9)

where c 11 , c 12 and c 44 are the elastic stiffnesses of the material and

E i j = 1 2 ( ∂ i u j + ∂ j u i ) . (10)

Here u i is the i^{th} Cartesian component of the displacement vector where the index i = 1 , 2 , 3 corresponds to the x , y , z coordinates respectively, and ∂ j indicates partial derivative with respect to the j^{th} coordinate. The quantities u i , σ i j , E i j are defined separately in the film (F) and substrate (S). Since mechanical equilibrium exists within the film and the substrate,

∂ j σ i j = 0 in F , S . (11)

Upon substituting the formulas for stress and strain in these equations we obtain Navier’s equations for the equilibrium displacements, which is valid both in the substrate and film:

c 11 ∂ 2 u 1 ∂ x 2 + ( c 12 + c 44 ) ∂ 2 u 2 ∂ x ∂ y + c 44 ∂ 2 u 1 ∂ y 2 + ( c 12 + c 44 ) ∂ 2 u 3 ∂ x ∂ z + c 44 ∂ 2 u 1 ∂ z 2 = 0 (12)

c 44 ∂ 2 u 2 ∂ x 2 + ( c 12 + c 44 ) ∂ 2 u 1 ∂ x ∂ y + c 11 ∂ 2 u 2 ∂ y 2 + ( c 12 + c 44 ) ∂ 2 u 3 ∂ y ∂ z + c 44 ∂ 2 u 2 ∂ z 2 = 0 (13)

c 44 ∂ 2 u 3 ∂ x 2 + ( c 12 + c 44 ) ∂ 2 u 1 ∂ x ∂ z + c 44 ∂ 2 u 3 ∂ y 2 + ( c 12 + c 44 ) ∂ 2 u 2 ∂ y ∂ z + c 11 ∂ 2 u 3 ∂ z 2 = 0 (14)

The stress balance boundary conditions at the film free surface,

σ i j F n ^ j = 0 on z = h ( x , y , t ) (15)

where

n ^ = ( − h x , − h y , 1 ) 1 + h x 2 + h y 2 (16)

is the unit normal to the film surface and at the film-substrate interface read

σ i j f n ^ j − σ i j s n ^ j = 0 on z = 0 (17)

The substrate is taken to be semi-infinite, and so the strains vanish far beneath the film,

E i j S → 0 as z → − ∞ . (18)

Finally on z = 0 (the film/substrate interface), continuity of displacement taking into account the lattice mismatch ò is

u i F = u i S + ϵ [ x y 0 ] . (19)

The evolution equation is given by surface diffusion in response to a chemical potential μ,

∂ h ∂ t = D 1 + | ∇ h | 2 ∇ S 2 ( μ ) (20)

where ∇ S 2 the surface Laplacian,

∇ S 2 = 1 1 + h x 2 + h y 2 [ ( 1 + h y 2 ) ∂ x 2 − 2 h x h y ∂ x ∂ y + ( 1 + h x 2 ) ∂ y 2 − ( 1 + h y 2 ) h x x − 2 h x h y h x y + ( 1 + h x 2 ) h y y 1 + h x 2 + h y 2 ( h x ∂ x + h y ∂ y ) ] (21)

D is a constant related to surface diffusion, and the surface chemical potential is

μ = E + γ κ + ω ( h ) (22)

where E is the elastic energy density, γ κ represents the surface energy, and ω ( h ) is the wetting energy. In the above,

E = 1 2 σ i j F E i j F on z = h ( x , y , t ) (23)

and the curvature of the film κ is given by

κ = − ( 1 + h y 2 ) h x x − 2 h x h y h x y + ( 1 + h x 2 ) h y y ( 1 + h x 2 + h y 2 ) 3 / 2 . (24)

For the wetting energy ω ( h ) we use the two-layer wetting model where the surface energy depends on the film thickness according to

γ ( h ) = γ F + ( γ S − γ F ) e − h / δ w (25)

The model for the wetting energy ω ( h ) is from [

γ ( h ) = 1 2 ( γ F + γ S ) + 1 2 ( γ F − γ S ) f ( h / δ ) , (26)

which gives the wetting term,

ω ( h ) = n y γ ′ ( h ) (27)

where

f ( h / δ ) = 2 π arctan ( h / δ ) (28)

and

Δ γ = γ S − γ F . (29)

Hence we get

ω ( h ) = − 1 1 + | ∇ h | 2 Δ γ π δ δ 2 + h 2 . (30)

Equation (20) is a non-linear moving boundary problem coupled to partial differential equations for the elasticity problem (12)-(19).

The governing equations in Section 2 describe the stress state and surface evolution of an epitaxially strained film. They have a basic-state solution corresponding to a completely relaxed, stress-free substrate,

u ¯ i s = 0 , σ ¯ i j s = 0 for i , j = 1 , 2 , 3 (31)

and a planar film with spatially uniform stress and strain,

u ¯ 1 f = ϵ x , u ¯ 2 f = ϵ y , u ¯ 3 f = ( − 2 c 12 ϵ c 11 ) z (32)

σ ¯ 11 f = σ ¯ 22 f = M 001 ϵ , σ ¯ 33 f = 0 (33)

where

M 001 = ( c 11 + c 12 − 2 c 12 2 c 11 ) (34)

which is the biaxial modulus in the plane with normal in the (001) material direction (in our case (001) material direction is the direction parallel to the z-axis).

The total elastic energy store in the film due to epitaxial and wetting stresses is

E 0 = 1 2 σ ¯ i j f E ¯ i j f = ϵ 2 M 001 (35)

In Section 5 we perform a linear stability analysis of this basic state of an epitaxial film.

Here we derive the evolution equation based on approximation that wavelength of surface undulations is large compared to the characteristic film thickness H 0 . Define

α = H 0 l ≪ 1 (36)

where l is characteristic length scale in ( x , y ) . Let us use the following scalings:

h = α H l x = l X y = l Y z = α l Z t = τ T u i ( x , y , z ) = l U i ( X , Y , Z ) δ = l δ ˜ } (37)

Next let us write the elastic stresses, strains and elastic energy density as expansions in α:

σ i j = ( σ i j ) 0 + α ( σ i j ) 1 + α 2 ( σ i j ) 2 + ⋯ (38)

E i j = ( E i j ) 0 + α ( E i j ) 1 + α 2 ( E i j ) 2 + ⋯ (39)

E = E 0 + α E 1 + α 2 E 2 + ⋯ (40)

Expanding the other quantities in (9) we obtain

κ = − 1 l ( α ( H X X + H Y Y ) + α 3 ( H Y 2 H X X − 2 H X H Y H X Y + H X 2 H Y Y − 3 / 2 | ∇ H | 2 ( H X X + H Y Y ) ) ) + O ( α 5 ) (41)

and the surface Laplacian has the expansion,

∇ S 2 = 1 l 2 [ ( ∂ X 2 + ∂ Y 2 ) + α 2 ( H Y 2 ∂ X 2 − 2 H X H Y ∂ X ∂ Y + H X 2 ∂ Y 2 + ( H X X + H Y Y ) ( H X ∂ X + H Y ∂ Y ) − | ∇ H | 2 ( ∂ X 2 + ∂ Y 2 ) ) ] + O ( α 4 ) . (42)

We choose the time scale τ as

τ = l 4 D γ (43)

and length scale l as

l = γ E 0 . (44)

We also assume that the characteristic film thickness is much larger than the wetting layer thickness, so

α ≫ δ ˜ . (45)

Hence (30) can be written as:

ω ( α l H ) ~ − Δ γ l π δ ˜ α 2 H 2 ( 1 − 1 2 α 2 | ∇ H | 2 − δ ˜ 2 α 2 H 2 + ⋯ ) . (46)

To balance the wetting energy term O ( δ ˜ / α 2 ) with the surface energy term O ( α ) in (20) we choose

δ ˜ = O ( α 3 ) (47)

and define

δ ˜ = δ * α 3 (48)

where δ * = O ( 1 ) . Then defining

ω ˜ = 1 α l γ ω ( α l H ) , (49)

we obtain

ω ˜ = ω ˜ 0 + α 2 ω ˜ 2 + O ( α 4 ) (50)

where

ω ˜ 0 = − Δ γ π γ δ * H 2 (51)

ω ˜ 2 = 1 2 − Δ γ π γ δ * H 2 | ∇ H | 2 . (52)

We substitute these expansions into the evolution Equation (20) to obtain

∂ H ∂ T = ( ∇ 2 ) ( E 1 − ∇ 2 H + ω ˜ 0 ) + α ( ∇ 2 ) E 2 + O ( α 2 ) (53)

where

E ˜ 1 = l E γ = E 1 E 0 (54)

and

E ˜ 2 = l E 2 γ = E 2 E 0 . (55)

Equation (53) is the thin film evolution equation. It depends on elastic response through E ˜ 1 and E ˜ 2 , which we now determine.

Now we find E ˜ 1 by solving the elasticity problem. Using scalings given in (37), (12)-(14) can be written as:

c 11 ∂ 2 U 1 ∂ X 2 + c 44 ∂ 2 U 1 ∂ Y 2 + c 44 1 α 2 ∂ 2 U 1 ∂ Z 2 + ( c 12 + c 44 ) ∂ 2 U 2 ∂ X ∂ Y + ( c 12 + c 44 ) 1 α ∂ 2 U 3 ∂ X ∂ Z = 0 , (56)

c 44 ∂ 2 U 2 ∂ X 2 + c 11 ∂ 2 U 2 ∂ Y 2 + c 44 1 α 2 ∂ 2 U 2 ∂ Z 2 + ( c 12 + c 44 ) ∂ 2 U 1 ∂ X ∂ Y + ( c 12 + c 44 ) 1 α ∂ 2 U 3 ∂ Y ∂ Z = 0 , (57)

The same way we can write stress and strain using (37) and from the boundary condition (15) we obtain:

on

We proceed by finding elasticity solutions in the film satisfying the boundary condition on

Now let us expand the displacements U in a,

We substitute (62) in (56)-(58) and compare by order in a.

At

with the boundary conditions,

We expect the

and solving (63) and using (64) we obtain

Solving the

Using

and

The same way using the

And the

and

Returning to our expansion for

and

The functions

The elasticity Equations (12)-(14) and far field condition (18) become

and

Using the Fourier transform to define

Then using (82) we write (78)-(80) as

and

in

We expand displacement in the substrate using powers in a,

with

Using (87) and the solution for

Using (86), (89) and (19), an

The

where

using (88) and (90)

The second boundary condition is (17). At

and

which gives us a linear system of equations for

and

Solving (97), (98) and (99) we obtain

Hence the Fourier transform

where

Hence in nondimensional form,

Note that the constant E contains the interaction of the elastic response of the anisotropic film and isotropic substrate.

The evolution equation at

where

with the strength of the wetting energy quantified by

Equation (106) contains the dominant effects for thin films, derived in a self-consistent thin film approximation. The three terms on the right hand side, correspond to elastic energy (

To determine the stability of planar films of thickness

We consider

When the film wets the substrate,

In

which every film thickness is stable,

and there is a critical wave number,

The evolution equation thus has the property that sufficiently thin films are stabilized by the wetting effect, but thicker films are unstable to the stress driven morphological instability.

We derived a self-contained evolution equation where the film thickness is smaller than wavelength of surface variations. Our evolution equation includes effects of anisotropic elastic constants for cubic symmetry in the film and isotropic elastic constants in the substrate, isotropic surface energy, and wetting energy. This evolution equation possesses steady state solutions corresponding to island formation, and is a possible candidate for use in large scale simulations of island systems.

Sincere thanks to the members of JAMP for their professional performance. And W. Tekalign would like to thank his Ph.D. adviser Dr. Brian J. Spencer for his valuable inputs and support throughout the years.

Tekalign, W. and Atena, A. (2018) Thin Film Evolution Equation for a Strained Anisotropic Solid Film on a Deformable Isotropic Substrate. Journal of Applied Mathematics and Physics, 6, 864-879. https://doi.org/10.4236/jamp.2018.64074