Letter pubs.acs.org/NanoLett
Comment on “Thermodynamic Self-Limiting Growth of Heteroepitaxial Islands Induced by Nonlinear Elastic Effect” István Daruka* Johannes Kepler University, Institute of Semiconductor and Solid State Physics, Altenbergerstrasse 69, A-4040 Linz, Austria
Nano Lett. 2016, 16 (6), 3919−3924, DOI: 10.1021/acs.nanolett.6b01525 Nano Lett. 2017, 17, DOI: 10.1021/acs.nanolett.7b00388 ABSTRACT: Making use of incontrovertible scaling arguments, here we demonstrate that the main findings of the recent Nano Letter paper Hu et al. (Hu, H.; Niu, X.; Liu, F. Nano Lett. 2016, 16, 3919) are incorrect and unphysical, that is, there is no selflimiting island growth due to nonlinear elastic effects, as claimed by Hu et al. We also note that the key concept of the paper by Hu et. al., that is, the island height dependence of the elastic strain field has already been published 10 years ago by Zinovyev, V. A.; Vastola, G.; Montalenti, F.; Miglio, L. Surf. Sci. 2006, 600, 4777, offering a correct and elaborate treatment even in a more generic sense, including also nonlinear elasticity effects consistently.
T
he recent Letter by Hu et al.1 reports on a self-limiting growth model of strained islands due to some nonlinear elastic effects. Here we show the model1 is built upon an inconsistent assumption, that is, the included nonlinear strain approximation is incorrect, rendering the main findings of the paper unphysical. In order to improve upon the already established shallow slope approximation, 3 Hu et al.1 include a nonlinear perturbation of the strain field
εHu(z) = ε0(1 − τ0z)
semi-infinite substrate (material B) and for simplicity we place the island at the origo. We denote the exact analytic solution for the corresponding three-dimensional nonlinear strain field ε1,ij(x,y,z). Then, the full, exact elastic relaxation energy can be written as Erel,1 =
where Cijkl is the elasticity matrix and e0(x,y,z) is the elastic energy density of the reference state. Next, we consider a second system, another three-dimensional strained island with the same geometry (material A), situated on top of a semi-infinite substrate (material B) but being scaled up in each direction by a linear factor f, that is, rendering x′ = f x, y′ = f y, and z′ = fz. Because of geometric similarity, the exact strain field in this second system will simply be
and claims that the such approximated elastic relaxation energy of a strained island scales as (2)
where the linear term represents the contribution of the shallow slope approximation and the V4/3 term is due to the included nonlinear perturbation of the strain field εHu (τ0, τ are considered constants and η is an island geometry-dependent coefficient). Furthermore, Hu et al.1 find that the latter V4/3 term gives rise to nonlinear elastic effects, claiming some novel, counterintuitive self-limiting growth of strained heteroepitaxial islands. It is important to emphasize that Hu et al.1 use the framework of linear elasticity theory4 throughout the paper and do not go beyond. We note that only the consideration of the height-dependent strain field εHu(z) renders some nonlinear strain effects to the problem, but it is treated fully in the framework of linear elasticity. This important distinction is not easily accessible from the text of the commented paper.1 As all the argumentations remain within the framework of linear elasticity, it is not the aim of this Comment to introduce and discuss the corresponding nonlinear elasticity frames. Instead, we suggest the reader to refer to ref 5. Here we argue that within the framework of linear elasticity theory,4 eq 2 can depend only linearly on the island volume V. In order to demonstrate that exactly, we assume a threedimensional strained island (material A) situated on top of a © XXXX American Chemical Society
∫ {Cijklε1,ij(x , y , z)ε1,kl(x , y , z) − e0(x , y , z)}dx dy dz (3)
(1)
Erel,Hu = −βV + τηV 4/3
1 2
ε2, ij(fx , fy, fz) = ε1, ij(x , y , z)
(4)
In this second case, the full (exact) elastic relaxation energy will be 1 {Cijklε2, ij(x′, y′, z′)ε2, kl(x′, y′, z′) − e0(x′, y′, z′)}dx′ dy′ dz′ 2 1 = {Cijklε1, ij(x , y , z)ε1, kl(x , y , z) − e0(x , y , z)}dx′ dy′ dz′ 2 1 = f 3 {Cijklε1, ij(x , y , z)ε1, kl(x , y , z) − e0(x , y , z)}dx dy dz 2
Erel,2 =
∫
∫
∫
= f 3 Erel,1
(5)
Now, if we introduce the elastic relaxation energy density erel,1 = Erel,1/V1 (for the first island, island volume V1), then we obtain Received: September 29, 2016 Revised: November 17, 2016
A
DOI: 10.1021/acs.nanolett.6b04086 Nano Lett. XXXX, XXX, XXX−XXX
Nano Letters Erel,2 = f 3 Erel,1 =
Erel,1 V1
V1f 3 = erel,1V2
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ACKNOWLEDGMENTS The author thanks the very constructive remarks of the two anonymous Reviewers, contributing towards the improvement of this Comment. This work was supported by the Austrian Science Funds (FWF), Project P28185.
(6)
As the linear scaling factor f can be arbitrary, we arrive at
Erel = erelV
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(7)
that is, the exact elastic relaxation energy of the island scales linearly with the island volume V if the island geometry is fixed. It is important to emphasize that this is an exact, incontrovertible result, in contrast to the approximation in ref 1. (In the above, just like in ref 1, we include no wetting layer and any contributions arising from possible surface stress discontinuities associated with island edges are also neglected.) Next, we note that for a fixed island shape, due to the above geometric similarity argument, that is, by scaling the island size linearly, in order to maintain the similarity relation eq 4 of the corresponding exact strain fields, τ0 (and τ) should scale inversely with the physical island height z0, thus also inversely with V1/3 (∼z0), that is, τ0 = τ0,corr(ϑ)/z0 (and τ = τcorr(ϑ) V−1/3). This means that eq 1 should be corrected as ⎛ z⎞ εcorr(z) = ε0⎜1 − τ0,corr(ϑ) ⎟ z0 ⎠ ⎝
Letter
REFERENCES
(1) Hu, H.; Niu, X.; Liu, F. Nano Lett. 2016, 16, 3919. (2) Zinovyev, V. A.; Vastola, G.; Montalenti, F.; Miglio, L. Surf. Sci. 2006, 600, 4777. (3) Tersoff, J.; Tromp, R. M. Phys. Rev. Lett. 1993, 70, 2782. (4) Landau, L. D; Lifshitz, E. M. Theory of Elasticity; Pergamon Press plc, 1959. (5) Ogden, R. W. Non-Linear Elastic Deformations; Courier Corporation, 1997.
(8)
Intuitively it means that for a larger island the actual physical height at which the elastic energy density gets reduced to a certain percentage will be proportionally larger for a fixed island shape. This key modification will naturally rescale the V4/3 dependence in ref 1 such that Erel,corr = − βV + τηV 4/3 = − βV + τcorr(ϑ)ηV −1/3V 4/3 = − (β − τcorr(ϑ)η)V
(9)
This linear dependence on the island volume V is in full agreement with the above geometrical-similarity-based scaling argument, and it also implies that there will be no V4/3 term in eq 2 as claimed by ref 1. Furthermore, we note that even if the commented approach1 was correct, it would not offer anything qualitatively new, as such nonlinear strain effects, that is, the correct consideration of the suggested island height dependence of the elastic strain field ε(z) has already been published 10 years ago2 in an even more generic sense and has also been cited by the currently commented paper1 itself. The elaborate treatment of ref 2 offers in fact a significant improvement upon the prevailing shallow slope approximation3 and includes also nonlinear elasticity effects consistently by performing detailed atomistic simulations. In summary, making use of the above exact, incontrovertible scaling arguments, and also considering the pertaining results in ref 2, we find that there will not be any related self-limiting island growth effects, rendering the main findings of ref 1 incorrect and unphysical.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
István Daruka: 0000-0002-8655-2660 Notes
The author declares no competing financial interest. B
DOI: 10.1021/acs.nanolett.6b04086 Nano Lett. XXXX, XXX, XXX−XXX
