Wondimu T. Tekalign, Agegnehu Atena
College of Engineering Technology and Computing, Savannah State University, Savannah, GA, USA.
DOI: 10.4236/jamp.2026.1410207   PDF    HTML   XML   1 Downloads   32 Views   Citations

Abstract

We consider a continuum model for the evolution of an epitaxially strained, dislocation-free thin solid film on a deformable substrate. In the absence of vapor deposition, we incorporate isotropic surface and strain energies along with a two-layer exponential wetting model. Applying a thin-film approximation, we successfully derive the corresponding evolution equation.

Keywords

Thin Films, Evolution Equation, Longwave Approximation

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1. Introduction

Heteroepitaxial thin films are formed by depositing a single-crystal layer onto a different single-crystal substrate that shares the same crystal structure but possesses a unique lattice spacing. These films typically vary in thickness from several nanometers to a few micrometers. When subjected to epitaxial strain, they often spontaneously organize into periodic, nanoscale island structures known as quantum dots. Owing to their exceptional electronic, optical, and magnetic characteristics, these islands are widely utilized in quantum computing, data storage, and advanced optoelectronic systems [1]-[4]. To engineer highly reliable and durable devices, it is critical to understand how specific growth mechanisms govern the structural evolution of these thin layers.

Because these islands are only a few nanometers in size, they are notoriously difficult to fabricate using conventional lithographic methods. A highly promising alternative is to generate them via the Stranski-Krastanow growth mode, a process in which an initially planar film undergoes a morphological instability [5]-[8]. This type of surface roughening—driven by elastic strain and the lattice mismatch between the film and substrate—has been a subject of extensive research [9] [10]. At its core, the resulting shape transition is dictated by a competitive trade-off between strain energy and surface energy [7] [11]-[15]. Furthermore, theoretical investigations have established that the planar surface of a stressed solid film is inherently unstable, making the formation of surface undulations energetically favorable [16]-[18].

Within a thin film, cusp formation is suppressed as the moving surface approaches the film-substrate interface. The unique stress fields and differing surface energies of the film and substrate profoundly influence the surface morphology. Furthermore, provided the wetting criterion is satisfied, the film-substrate interface is prevented from being exposed, resulting instead in the formation of Stranski-Krastanov islands [19]-[22]. Numerous researchers have subsequently investigated the steady-state shapes of these islands [12] [23]-[26]. Although the theoretical frameworks and underlying modeling concepts are well advanced, implementing these models via large-scale numerical simulations remains challenging. The primary hurdle is that surface morphology dynamics are strongly coupled to the elastic strain within the system. Consequently, dynamic models must solve the elasticity equations across both the film and the substrate at every single time step—a requirement that creates severe computational bottleneck constraints for three-dimensional problems. Until recently, simulations incorporating the full elasticity problem were restricted to just one or a few islands [27]. Among recent large-scale studies, [28] examined a large population of islands by assuming identical elastic constants for both the film and substrate to simplify the mechanics. Similarly, [29] simulated numerous islands by applying a small-slope approximation restricted to the special case of an elastically rigid substrate. To address these limitations, our approach introduces a systematic, approximate solution to the elasticity problem. This method remains highly valid for thin films while successfully accommodating the general case where the film and substrate possess distinct elastic constants. The foundational thin-film methodology used here originates from the lubrication theory of viscous fluids [30]. By eliminating the need to numerically solve the complete three-dimensional elasticity equations, this formulation provides a viable path toward executing large-scale simulations. Prior work in [31] explored a comparable framework specifically for a rigid substrate in the absence of vapor deposition. That study demonstrated that a planar film becomes unstable once it exceeds a critical thickness, with the instability characterized by long waves at that threshold. In [32], a long-wave approximation was deployed alongside vapor deposition—but without wetting energy considerations—to model the emergence of faceted pyramidal structures. Additionally, [33] derived a nonlinear evolution equation for stress-driven morphological instability governed by surface diffusion in two-dimensional stressed solids without deposition. By obtaining a nonlinear approximation of the elastic energy density and applying numerical methods to the evolution equation, they observed a surface that evolves from small perturbations toward a cusp-like morphology. Our work distinguishes itself from these previous methodologies by directly evaluating a coupled film-substrate system where the film and substrate can feature completely different elastic properties.

2. Model for Film Growth

Governing Equations

Figure 1 illustrates a schematic layout of the surface evolution. We build upon the model introduced by [16], which describes how surface diffusion drives the evolution of a strained film in response to elastic strain and surface energies. This formulation assumes an “atomically rough” surface. While alternative models utilize surface steps to derive the governing evolution equations [34], we restrict our focus here to the simpler case of film annealing without vapor deposition. Geometrically, the film occupies the region $0<z<h\left(x,y,t\right)$ , the substrate spans $z<0$ , and the vapor phase resides in $z>h\left(x,y,t\right)$ under zero pressure.

Figure 1. As time increases the originally flat film evolves to form islands.

The stress in the film and the substrate are governed by isotropic linear elasticity,

${\sigma }_{ij}=\frac{2\mu \nu }{1-2\nu }{\delta }_{ij}{E}_{kk}+2\mu E_{ij}$ (1)

where $E_{ij}$ is the linear elasticity strain tensor given by

$E_{ij}=\frac{1}{2}\left({\partial }_i{u}_j+{\partial }_j{u}_i\right).$ (2)

The variables in the governing equations are given in the following Table 1. Here $u_i$ is the $i^{\text{th}}$ Cartesian component of the displacement vector where the index $i=1,2,3$ corresponds to the $x,y,z$ coordinates respectively, and ${\partial }_j$ indicates partial derivative with respect to the $j^{\text{th}}$ coordinate. Repeated indices imply summation from 1 to 3. The quantities $u_i,{\sigma }_{ij},E_{ij},\mu ,\nu$ are defined separately in the film (F) and substrate (S). Since mechanical equilibrium exists within the film and the substrate,

Table 1. List of symbols.

symbol	meaning
$\mu$	elastic shear modulus
$\nu$	Poisson’s ratio
${\delta }_{ij}$	Kronecker’s delta

${\partial }_j{\sigma }_{ij}=0\text{in}\text{F},\text{S}.$ (3)

Upon substituting the formulas for stress and strain in this equation we obtain Navier’s equations

$\left(1-2\nu \right){\partial }_k^2{u}_i+{\partial }_k{\partial }_i{u}_k=0\text{in}\text{F},\text{S}.$ (4)

On $z=h\left(x,y,t\right)$ the force on the film surface (the tractions due to the stress in the direction of the normal $\widehat{n}$ ) is equal to the pressure in the vapor,

${\sigma }_{ij}^F{\widehat{n}}_j=0$ (5)

where

$\widehat{n}=\frac{\left(-h_x,-h_y,1\right)}{\sqrt{1+h_x^2+h_y^2}}.$

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

$u_i^S\to 0\text{as}z\to -\infty .$ (6)

On $z=0$ (the film/substrate interface), the misfit condition gives

$u_i^F=u_i^S+ϵ\left[\begin{array}{c}x\\ y\\ 0\end{array}\right]$ (7)

where $ϵ$ is the misfit strain in the film, induced by the lattice matching of the film to the substrate, characterized by

$ϵ=\left(a_f-a_s\right)/a_s$ (8)

where $a_f$ and $a_s$ are the lattice spacings of the film and the substrate, respectively. Finally, a force balance on $z=0$ gives

${\sigma }_{ij}^F\widehat{z}={\sigma }_{ij}^S\widehat{z}$ (9)

where $\widehat{z}$ is unit vector in the $z$ direction.

The evolution equation for the surface $z=h\left(x,y,t\right)$ is determined by surface diffusion in response to a chemical potential $ℳ$ ,

$\frac{\partial h}{\partial t}=D\sqrt{1+{\left|\nabla h\right|}^2}{\nabla }_S^2\left(ℳ\right)$ (10)

where ${\nabla }_S^2$ is the surface Laplacian,

$\begin{array}{c}{\nabla }_S^2=\frac{1}{1+h_x^2+h_y^2}[\left(1+h_y^2\right){\partial }_x^2-2h_x{h}_y{\partial }_x{\partial }_y+\left(1+h_x^2\right){\partial }_y^2\begin{array}{c}{}^{}\\ {{\displaystyle }}_{}^{}\end{array}\\ -\frac{\left(1+h_y^2\right)h_{xx}-2h_x{h}_y{h}_{xy}+\left(1+h_x^2\right)h_{yy}}{1+h_x^2+h_y^2}\left(h_x{\partial }_x+h_y{\partial }_y\right)]\end{array}$ (11)

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

$ℳ=ℰ+\gamma \kappa +\omega \left(h\right)$ (12)

where $ℰ$ is the elastic energy density, $\gamma \kappa$ represents the surface energy, and $\omega \left(h\right)$ is the wetting energy. In the above,

$ℰ=\frac{1}{2}{\sigma }_{ij}^F{E}_{ij}^F\text{on}z=h\left(x,y,t\right)$ (13)

and the curvature of the film $\kappa$ is given by

$\kappa =-\frac{\left(1+h_y^2\right)h_{xx}-2h_x{h}_y{h}_{xy}+\left(1+h_x^2\right)h_{yy}}{{\left(1+h_x^2+h_y^2\right)}^{3/2}}.$ (14)

3. Model for the Wetting Energy

To model the wetting energy $\omega \left(h\right)$ we use a two layer exponential form, based on a surface energy which depends on the film thickness and undergoes a rapid transition from ${\gamma }_F$ to ${\gamma }_S$ over a length scale $\delta$ :

$\gamma \left(h\right)=\frac{1}{2}\left({\gamma }_F+{\gamma }_S\right)+\frac{1}{2}\left({\gamma }_F-{\gamma }_S\right)f\left(h/\delta \right),$ (15)

which gives the wetting term,

$\omega \left(h\right)={\widehat{n}}_z{\gamma }^{\prime }\left(h\right)$ (16)

where ${\widehat{n}}_z$ is the $z$ component of $\widehat{n}$ ,

$f\left(h/\delta \right)=1-{\text{e}}^{-h/\delta }$ (17)

and

$\Delta \gamma ={\gamma }_S-{\gamma }_F.$

Hence we get

$\omega \left(h\right)=\frac{1}{\sqrt{1+{\left|\nabla h\right|}^2}}\frac{△\gamma }{2\delta }{\text{e}}^{-h/\delta }.$ (18)

Equation (10) is a non-linear moving boundary problem coupled to partial differential equations for the elasticity problem (4)-(9).

4. Thin-Film Approximation and Derivations

Following the scalings, notations, and derivations presented in [35], the nondimensionalized wetting energy density is obtained as:

$\omega \left(\alpha lH\right)~\frac{-\Delta \gamma }{l\pi }\frac{\tilde{\delta }}{{\alpha }^2{H}^2}\left(1-\frac{1}{2}{\alpha }^2{\left|\nabla H\right|}^2-\frac{{\tilde{\delta }}^2}{{\alpha }^2{H}^2}+\cdots \right).$ (19)

To balance the wetting energy term $\mathcal{O}\left(\tilde{\delta }/{\alpha }^2\right)$ with the surface energy term $\mathcal{O}\left(\alpha \right)$ in (12) we choose

$\tilde{\delta }=\mathcal{O}\left({\alpha }^3\right)$

and define

$\tilde{\delta }={\delta }^{\text{*}}{\alpha }^3$

where ${\delta }^{\text{*}}=\mathcal{O}\left(1\right)$ . Then defining

$\tilde{\omega }=\frac{1}{\alpha }\frac{l}{\gamma }\omega \left(\alpha lH\right),$ (20)

we obtain

$\tilde{\omega }={\tilde{\omega }}_0+{\alpha }^2{\tilde{\omega }}_2+\mathcal{O}\left({\alpha }^4\right)$ (21)

where

${\tilde{\omega }}_0=\frac{-\Delta \gamma }{\pi \gamma }\frac{{\delta }^{\text{*}}}{H^2}$

${\tilde{\omega }}_2=\frac{1}{2}\frac{-\Delta \gamma }{\pi \gamma }\frac{{\delta }^{\text{*}}}{H^2}{\left|\nabla H\right|}^2.$

We substitute these expansions into the evolution Equation (10) to obtain

$\frac{\partial H}{\partial T}={\nabla }^2\left({\widetilde{ℰ}}_1-{\nabla }^{2}H+{\tilde{\omega }}_0\right)+\alpha {\nabla }^2{\widetilde{ℰ}}_2+\mathcal{O}\left({\alpha }^2\right)$ (22)

where

${\widetilde{ℰ}}_1=\frac{l{ℰ}_1}{\gamma }=\frac{{ℰ}_1}{{ℰ}_0}$

and

${\widetilde{ℰ}}_2=\frac{l{ℰ}_2}{\gamma }=\frac{{ℰ}_2}{{ℰ}_0}.$

Equation (22) is the thin film evolution equation. It depends on elastic response through ${\widetilde{ℰ}}_1$ and ${\widetilde{ℰ}}_2$ , which we now determine.

Evolution Equation

Returning to Equation (22) using the expansions derived above we get

$\begin{array}{l}\frac{\partial H}{\partial T}={\nabla }^2\left({\widetilde{ℰ}}_1-{\nabla }^{2}H+{\tilde{\omega }}_0\right)+\alpha {\nabla }^2{\widetilde{ℰ}}_2\\ {\alpha }^2\left[{\nabla }^2\left({\widetilde{ℰ}}_3-H_Y^2{H}_{XX}+2H_X{H}_Y{H}_{XY}-H_X^2{H}_{YY}+\frac{3}{2}{\left|\nabla H\right|}^2{\nabla }^{2}H+{\tilde{\omega }}_2\right)+\cdots \right]+\mathcal{O}\left({\alpha }^3\right)\end{array}$

where

${\widetilde{ℰ}}_j=F^{-1}\left[{\widehat{\widetilde{ℰ}}}_j\right]$

This evolution equation is derived in a self-consistent thin film approximation. It can be used to characterize pattern formation in thin films with order of $\alpha$ we are interested in.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References

[1]	Scheffler, M. and Zimmermann, R. (1996) 23rd International Conference on Physics of Semiconductors: Berlin, Germany July 21-26, 1996. Germany World Scientific Publishing.
[2]	Chen, M. and Porod, W. (1995) Design of Gate-Confined Quantum-Dot Structures in the Few-Electron Regime. Journal of Applied Physics, 78, 1050-1057.[CrossRef]
[3]	Gao, H. and Nix, W.D. (1999) Surface Roughening of Heteroepitaxial Thin Films. Annual Review of Materials Science, 29, 173-209.[CrossRef]
[4]	Petroff, P.M., Lorke, A. and Imamoglu, A. (2001) Epitaxially Self-Assembled Quantum Dots. Physics Today, 54, 46-52.[CrossRef]
[5]	Asaro, R.J. and Tiller, W.A. (1972) Interface Morphology Development during Stress Corrosion Cracking: Part I. via Surface Diffusion. Metallurgical Transactions, 3, 1789-1796.[CrossRef]
[6]	Grinfeld, M.A. (1993) The Stress Driven “Rearrangement” Instability in Crystalline Films. Journal of Intelligent Material Systems and Structures, 4, 76-81.[CrossRef]
[7]	Srolovitz, D.J. (1989) On the Stability of Surfaces of Stressed Solids. Acta Metallurgica, 37, 621-625.[CrossRef]
[8]	Spencer, B.J., Voorhees, P.W. and Davis, S.H. (1991) Morphological Instability in Epitaxially Strained Dislocation-Free Solid Films. Physical Review Letters, 67, 3696-3699.[CrossRef] [PubMed]
[9]	Gao, H. (1994) Some General Properties of Stress-Driven Surface Evolution in a Heteroepitaxial Thin Film Structure. Journal of the Mechanics and Physics of Solids, 42, 741-772.[CrossRef]
[10]	Jonsdottir, F. and Freund, L.B. (1995) Equilibrium Surface Roughness of a Strained Epitaxial Film Due to Surface Diffusion Induced by Interface Misfit Dislocations. Mechanics of Materials, 20, 337-349.[CrossRef]
[11]	Daruka, I., Tersoff, J. and Barabási, A.L. (1999) Shape Transition in Growth of Strained Islands. Physical Review Letters, 82, 2753-2756.[CrossRef]
[12]	Spencer, B.J. and Tersoff, J. (1997) Equilibrium Shapes and Properties of Epitaxially Strained Islands. Physical Review Letters, 79, 4858-4861.[CrossRef]
[13]	Nix, W.D. (1989) Mechanical Properties of Thin Films. Metallurgical Transactions A, 20, 2217-2245.[CrossRef]
[14]	Gao, H. (1991) Stress Concentration at Slightly Undulating Surfaces. Journal of the Mechanics and Physics of Solids, 39, 443-458.[CrossRef]
[15]	Gao, H. (1991) A Boundary Perturbation Analysis for Elastic Inclusions and Interfaces. International Journal of Solids and Structures, 28, 703-725.[CrossRef]
[16]	Spencer, B.J., Voorhees, P.W. and Davis, S.H. (1993) Morphological Instability in Epitaxially Strained Dislocation-Free Solid Films: Linear Stability Theory. Journal of Applied Physics, 73, 4955-4970.[CrossRef]
[17]	Freund, L.B. and Jonsdottir, F. (1993) Instability of a Biaxially Stressed Thin Film on a Substrate Due to Material Diffusion over Its Free Surface. Journal of the Mechanics and Physics of Solids, 41, 1245-1264.[CrossRef]
[18]	Spencer, B.J. and Meiron, D.I. (1994) Nonlinear Evolution of the Stress-Driven Morphological Instability in a Two-Dimensional Semi-Infinite Solid. Acta Metallurgica et Materialia, 42, 3629-3641.[CrossRef]
[19]	Orr, B.G. and Snyder, C.W. (1992) A Model for Lattice-Mismatched Epitaxy: A Continuum View. MRS Proceedings, 280, Article No. 429.[CrossRef]
[20]	Ratsch, C. and Zangwill, A. (1993) Equilibrium Theory of the Stranski-Krastanov Epitaxial Morphology. Surface Science, 293, 123-131.[CrossRef]
[21]	Barabási, A. (1997) Self-Assembled Island Formation in Heteroepitaxial Growth. Applied Physics Letters, 70, 2565-2567.[CrossRef]
[22]	Albrecht, M., Christiansen, S., Michler, J., Dorsch, W., Strunk, H.P., Hansson, P.O., et al. (1995) Surface Ripples, Crosshatch Pattern, and Dislocation Formation: Cooperating Mechanisms in Lattice Mismatch Relaxation. Applied Physics Letters, 67, 1232-1234.[CrossRef]
[23]	Spencer, B.J. and Tersoff, J. (1995) Equilibrium Shapes of Small Strained Islands. MRS Proceedings, 399, Article No. 283.[CrossRef]
[24]	Kukta, R.V. and Freund, L.B. (1997) Minimum Energy Configuration of Epitaxial Material Clusters on a Lattice-Mismatched Substrate. Journal of the Mechanics and Physics of Solids, 45, 1835-1860.[CrossRef]
[25]	Rudin, C.D. and Spencer, B.J. (1999) Equilibrium Island Ridge Arrays in Strained Solid Films. Journal of Applied Physics, 86, 5530-5536.[CrossRef]
[26]	Shanahan, L.L. and Spencer, B.J. (2002) A Codimension-Two Free Boundary Problem for the Equilibrium Shapes of a Small Three-Dimensional Island in an Epitaxially Strained Solid Film. Interfaces and Free Boundaries, Mathematical Analysis, Computation and Applications, 4, 1-25.[CrossRef]
[27]	Zhang, Y.W. and Bower, A.F. (1999) Numerical Simulations of Island Formation in a Coherent Strained Epitaxial Thin Film System. Journal of the Mechanics and Physics of Solids, 47, 2273-2297.[CrossRef]
[28]	Liu, P., Zhang, Y.W. and Lu, C. (2003) Coarsening Kinetics of Heteroepitaxial Islands in Nucleationless Stranski-Krastanov Growth. Physical Review B, 68, Article ID: 035402.[CrossRef]
[29]	Golovin, A.A., Davis, S.H. and Voorhees, P.W. (2003) Self-Organization of Quantum Dots in Epitaxially Strained Solid Films. Physical Review E, 68, Article ID: 056203.[CrossRef] [PubMed]
[30]	Oron, A., Davis, S.H. and Bankoff, S.G. (1997) Long-Scale Evolution of Thin Liquid Films. Reviews of Modern Physics, 69, 931-980.[CrossRef]
[31]	Spencer, B.J., Davis, S.H. and Voorhees, P.W. (1993) Morphological Instability in Epitaxially Strained Dislocation-Free Solid Films: Nonlinear Evolution. Physical Review B, 47, 9760-9777.[CrossRef] [PubMed]
[32]	Savina, T.V., Golovin, A.A., Davis, S.H., Nepomnyashchy, A.A. and Voorhees, P.W. (2003) Faceting of a Growing Crystal Surface by Surface Diffusion. Physical Review E, 67, Article ID: 021606.[CrossRef] [PubMed]
[33]	Xiang, Y. and E, W. (2002) Nonlinear Evolution Equation for the Stress-Driven Morphological Instability. Journal of Applied Physics, 91, 9414-9422.[CrossRef]
[34]	Caflisch, R.E. and Li, B. (2003) Analysis of Island Dynamics in Epitaxial Growth of Thin Films. Multiscale Modeling & Simulation, 1, 150-171.[CrossRef]
[35]	Tekalign, W.T. and Spencer, B.J. (2007) Thin-Film Evolution Equation for a Strained Solid Film on a Deformable Substrate: Numerical Steady States. Journal of Applied Physics, 102, Article ID: 073503.[CrossRef]