^{1}, Dwaipayan Dasgupta

^{1}and Dimitrios Maroudas

^{1,a)}

### Abstract

We report a theoretical analysis on the surface morphological stability of a coherently strained thin film that has been grown epitaxially on a deformable substrate and is simultaneously subjected to an external electric field and a temperature gradient. Using well justified approximations, we develop a three-dimensional model for the surface morphological evolution of the thin film and conduct a linear stability analysis of the heteroepitaxial film's planar surface state. The effect of the simultaneous action of multiple external fields on the surface diffusional anisotropy tensor is accounted for. Various substrate types are considered, but emphasis is placed on a compliant substrate that has the ability to accommodate elastically some of the misfit strain in the film due to its lattice mismatch with the substrate. We derive the condition for the synergy or competition of the two externally applied fields and determine the optimal alignment of the external fields that minimizes the critical electric field-strength requirement for the stabilization of the planar film surface. We also examine the role of the temperature dependence of the thermophysical properties and show that the criticality condition for planar surface stabilization does not change when the Arrhenius temperature dependence of the surface diffusivity is considered. Our analysis shows that surface electromigration and thermomigration due to the simultaneous action of properly applied and sufficiently strong electric fields and thermal gradients, respectively, can inhibit Stranski-Krastanow-type instabilities and control the onset of island formation on epitaxial film surfaces.

This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under Award No. DE-FG02-07ER46407.

I. INTRODUCTION

II. THEORETICAL ANALYSIS

A. Height evolution equation

B. Electrostatic BVP

C. Elastostatic deformation BVP

D. Thermal BVP

E. Linear stability analysis

F. Temperature dependence of the thermophysical properties

III. RESULTS AND DISCUSSION

IV. CONCLUSIONS

### Key Topics

- Surface morphology
- 105.0
- Epitaxy
- 47.0
- Electric fields
- 38.0
- External field
- 31.0
- Elasticity
- 25.0

##### C23C

## Figures

Schematic depiction of a heteroepitaxial conducting thin film on a finite-thickness compliant substrate subjected to an external electric field E∞ and a thermal gradient . (a) Initial configuration of the heteroepitaxial film/substrate system with a planar film surface morphology. (b) Top view of the epitaxial thin film surface; the directions of the two externally applied fields are shown with respect to the FSDD. The directions in the schematic of (b) are chosen to aid visually in defining the misorientation angles, ϕE and ϕT . In the analysis, the applied field directions are taken to be generally arbitrary.

Schematic depiction of a heteroepitaxial conducting thin film on a finite-thickness compliant substrate subjected to an external electric field E∞ and a thermal gradient . (a) Initial configuration of the heteroepitaxial film/substrate system with a planar film surface morphology. (b) Top view of the epitaxial thin film surface; the directions of the two externally applied fields are shown with respect to the FSDD. The directions in the schematic of (b) are chosen to aid visually in defining the misorientation angles, ϕE and ϕT . In the analysis, the applied field directions are taken to be generally arbitrary.

Polar plots on the surface plane of the anisotropy function of the surface diffusivity tensor in the absence of any externally applied fields. The thin dashed line (circle) corresponds to isotropic surface diffusion, while the thick dashed lines mark the symmetry axes of the fcc film surface that correspond to fast surface diffusion directions. The number of symmetry axes is given by the parameter m of . (a) m = 1, (b) m = 2, and (c) m = 3. In all cases, the value of the anisotropy strength parameter of is A = 12.

Polar plots on the surface plane of the anisotropy function of the surface diffusivity tensor in the absence of any externally applied fields. The thin dashed line (circle) corresponds to isotropic surface diffusion, while the thick dashed lines mark the symmetry axes of the fcc film surface that correspond to fast surface diffusion directions. The number of symmetry axes is given by the parameter m of . (a) m = 1, (b) m = 2, and (c) m = 3. In all cases, the value of the anisotropy strength parameter of is A = 12.

Polar plots on the surface plane of the anisotropy function of the surface diffusivity tensor under the simultaneous action of the applied electric and thermal fields of Fig. 1 . The solid and dashed arrows indicate the directions of the applied electric and thermal fields, respectively. The anisotropy parameters are (a, d, g) m = 1, (b, e, h) m = 2, and (c, f, i) m = 3, while A = 12 in all cases (a)-(i). (a)–(c) Only an electric field is applied, i.e., , and . (d)–(f) Both an electric and a thermal field are applied with equal strengths, , , and . (g)–(i) Both an electric and a thermal field are applied with different strengths, , , and . In cases (d)-(i), the effective angle, ϕeff , is calculated from Eqs. (7) and (8) .

Polar plots on the surface plane of the anisotropy function of the surface diffusivity tensor under the simultaneous action of the applied electric and thermal fields of Fig. 1 . The solid and dashed arrows indicate the directions of the applied electric and thermal fields, respectively. The anisotropy parameters are (a, d, g) m = 1, (b, e, h) m = 2, and (c, f, i) m = 3, while A = 12 in all cases (a)-(i). (a)–(c) Only an electric field is applied, i.e., , and . (d)–(f) Both an electric and a thermal field are applied with equal strengths, , , and . (g)–(i) Both an electric and a thermal field are applied with different strengths, , , and . In cases (d)-(i), the effective angle, ϕeff , is calculated from Eqs. (7) and (8) .

(a) Heteroepitaxial film/substrate system configuration with a plane-wave perturbation introduced into the thin film's surface morphology; the amplitude of the perturbation has been magnified for clarity. (b) An example of alignment of the externally applied fields with the direction of the temperature gradient and the FSDD coinciding with the x-axis.

(a) Heteroepitaxial film/substrate system configuration with a plane-wave perturbation introduced into the thin film's surface morphology; the amplitude of the perturbation has been magnified for clarity. (b) An example of alignment of the externally applied fields with the direction of the temperature gradient and the FSDD coinciding with the x-axis.

Determination of the synergy or competition of the two externally applied fields for all their possible orientations on the surface. The surface plots (a), (c), and (e) show the dependence of the field synergy/competition parameter, , on the directions of the applied electric and thermal fields, as determined by the misorientation angles and , respectively; (a) m = 1, (c) m = 2, and (e) m = 3. The corresponding contour plots are shown in (b), (d), and (f) for m = 1, 2, and 3, respectively. In all cases, dark and light shading indicate negative and positive values of the parameter , respectively, i.e., competition and synergy, respectively, of the two externally applied fields. The values of the parameters used are b = 1, A = 12, and Θ = 1.

Determination of the synergy or competition of the two externally applied fields for all their possible orientations on the surface. The surface plots (a), (c), and (e) show the dependence of the field synergy/competition parameter, , on the directions of the applied electric and thermal fields, as determined by the misorientation angles and , respectively; (a) m = 1, (c) m = 2, and (e) m = 3. The corresponding contour plots are shown in (b), (d), and (f) for m = 1, 2, and 3, respectively. In all cases, dark and light shading indicate negative and positive values of the parameter , respectively, i.e., competition and synergy, respectively, of the two externally applied fields. The values of the parameters used are b = 1, A = 12, and Θ = 1.

Dispersion relations giving the dependence, according to Eq. (52b) , of the characteristic rate, ω, for the growth or decay of the perturbation from the planar film surface morphology on the dimensionless wave number of the perturbation for Ξ E = , , and , (dashed, solid, and dotted line, respectively), where is the dimensionless critical strength of the applied electric field for stabilization of the planar morphology of the epitaxial film surface. The values of the parameters used are Θ = 1, ϕT = 0, b = 1, and and the substrate is compliant.

Dispersion relations giving the dependence, according to Eq. (52b) , of the characteristic rate, ω, for the growth or decay of the perturbation from the planar film surface morphology on the dimensionless wave number of the perturbation for Ξ E = , , and , (dashed, solid, and dotted line, respectively), where is the dimensionless critical strength of the applied electric field for stabilization of the planar morphology of the epitaxial film surface. The values of the parameters used are Θ = 1, ϕT = 0, b = 1, and and the substrate is compliant.

Dependence of the dimensionless critical electric-field strength, , on the misorientation angle, , for a thermal gradient that is applied parallel to the Cartesian x-axis, , and at an angle with respect to the x-axis (inset). The values of the parameters used are Θ = 1, b = 1, and , and the substrate is compliant. In both cases, the minimum is exhibited at , determining an optimal direction for the application of the external electric field, E∞ .

Dependence of the dimensionless critical electric-field strength, , on the misorientation angle, , for a thermal gradient that is applied parallel to the Cartesian x-axis, , and at an angle with respect to the x-axis (inset). The values of the parameters used are Θ = 1, b = 1, and , and the substrate is compliant. In both cases, the minimum is exhibited at , determining an optimal direction for the application of the external electric field, E∞ .

Dependence of the dimensionless critical electric-field strength on the surface diffusional anisotropy strength A for crystallographic orientations of the film plane corresponding to symmetry parameter values m = 1, 2, and 3. The values of the parameters used are Θ = 1, b = 1, and and the substrate is compliant. The electric field is applied at angles , , and , i.e., their optimal values for surface stabilization for each crystallographic orientation.

Dependence of the dimensionless critical electric-field strength on the surface diffusional anisotropy strength A for crystallographic orientations of the film plane corresponding to symmetry parameter values m = 1, 2, and 3. The values of the parameters used are Θ = 1, b = 1, and and the substrate is compliant. The electric field is applied at angles , , and , i.e., their optimal values for surface stabilization for each crystallographic orientation.

Dependence of the range of unstable wave numbers, R, on the dimensionless electric-field strength Ξ E with (dashed lines) and without (solid lines) the simultaneous action of the thermal field for (a) an infinitely thick and rigid substrate (thick black lines) and a finite-thickness substrate with clamped onto a rigid holder (thin gray lines) and (b) for a thin compliant substrate with . The inset in (b) shows the R(Ξ E ) dependence for the optimal compliant-substrate case for . Thevalues of the parameters used are Θ = 1, b = 1, , and .

Dependence of the range of unstable wave numbers, R, on the dimensionless electric-field strength Ξ E with (dashed lines) and without (solid lines) the simultaneous action of the thermal field for (a) an infinitely thick and rigid substrate (thick black lines) and a finite-thickness substrate with clamped onto a rigid holder (thin gray lines) and (b) for a thin compliant substrate with . The inset in (b) shows the R(Ξ E ) dependence for the optimal compliant-substrate case for . Thevalues of the parameters used are Θ = 1, b = 1, , and .

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