Thermodynamic Self-Limiting Growth of Heteroepitaxial Islands

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Thermodynamic Self-limiting Growth of Heteroepitaxial Islands induced by Nonlinear Elastic Effect Hao Hu, Xiaobin Niu, and Feng Liu Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.6b01525 • Publication Date (Web): 20 May 2016 Downloaded from http://pubs.acs.org on May 23, 2016

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Thermodynamic Self-limiting Growth of Heteroepitaxial Islands induced by Nonlinear Elastic Effect Hao Hu1, Xiaobin Niu2, Feng Liu3,4,* 1

Frontier Institute of Science and Technology, Xi'an Jiaotong University, Xi'an 710054,

Shaanxi, China 2

State Key Laboratory of Electronic Thin Film and Integrated Devices, University of

Electronic Science and Technology of China, Chengdu 610054, China 3

Department of Materials Science and Engineering, University of Utah, Salt Lake City,

UT 84112, USA 4

Collaborative Innovation Center of Quantum Matter, Beijing 100084, China

ABSTRACT: We investigate nonlinear elastic effect (NLEF) on the growth of heteroepitaxial islands, a topic of both scientific and technological significance for their applications as quantum dots. We show that the NLEF induces a thermodynamic self-limiting growth mechanism that hinders the strain relaxation of coherent island beyond a maximum size, in contrast to indefinite strain relaxation with increasing island size in the linear elastic regime. This self-limiting growth effect shows a strong dependence on the island facet angle, which applies also to islands inside pits patterned in a substrate surface with an additional dependence on the pit inclination angle. Consequently, primary islands nucleate and grow first in the pits, and then secondary islands nucleate at the rim around the pits after the primary islands reach the self-limited maximum size. Our theory sheds new lights on understanding the heteroepitaxial island growth and explains a number of past and recent experimental observations.

ACS Paragon Plus Environment

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KEYWORDS:

quantum

dots,

nonlinear

elastic

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effect,

nanopatterned substrate


self-limiting

growth,

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Since first discovered in 19901, heteroepitaxially grown self-assembled 3D islands have drawn continued attention due to fundamental interest in understanding their complex growth behavior and their potential applications as quantum dots in nano electronic and optoelectronic devices.2-5 Growth of such islands is a strain-mediated and thermally activated nucleation process, characterized by random island positions and broad size distribution.6-8 However, by manipulating the strain field on the surface prior to growth, island nucleation can be directed and growth be controlled to achieve highly ordered island arrays with significantly improved size uniformity. The most successful examples include using multilayer growth with buried islands9,10 and patterned substrates with surface ridges, grooves, mesas and metal patterns11-16, and most recently pits17-24. Despite extensive experimental and theoretical studies, however, fundamental understanding of nucleation and growth of strained islands, especially on patterned substrates is far from complete. Our basic understanding has so far relied mostly on theoretical analyses based on continuum linear elastic theory25-32. Island formation is triggered by a competition between surface energy and strain energy: the former increases with island volume in the power of 2/3 and the latter decreases with island volume in the power of 1, so that strain relaxation always wins beyond a critical island size, as shown in Fig. 1. The linear-elastic theory has enjoyed tremendous success in explaining many experimental observations as well as guiding new experiments.6-32 However, they are in principle limited to islands with shallow facet angels (shallow-angle approximation25-32) and relatively small size, or nonlinear elastic effects should kick in. For example, within the linear elastic regime, thermodynamically the strain relaxation energy decreases continuously with the


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increasing island size, so that island would continue to grow indefinitely, while experimentally island sizes are limited before dislocation formation.

1,17,33-35

For this

reason, kinetic self-limiting growth mechanisms36-39 have been incorporated to explain the experiments. Furthermore, recent experiments have shown an intriguing behavior of island growth on a pit-patterned substrate. The pit inclination angle is effectively used to change the island positions20,21,23 as well as delay the onset of plastic relaxation.17 These observations cannot be explained by the linear elastic approach. In this Letter, we develop a theoretical model that goes beyond linear elasticity by including nonlinear elastic effects in calculating strain energy. We show that nonlinear strain relaxation gives rise to a thermodynamic self-limiting growth mechanism, which is unknown before. It may trigger dislocation formation within a coherent island beyond a maximum size that decreases with the increasing island facet angle, i.e., the steeper the island, the sooner the dislocation forms. Furthermore, the island growing in a pit depends strongly on both island facet angle and pit inclination angle, leading to complex growth behavior that are all consistent with experimental observations. Island growth on the surface will induce strained film thickness variation, and consequently the surface stress variation.25,29-32 In the linear elastic theory, the surface stress is proportional to the film thickness as well as the bulk strain in the film, i.e. σ(x,y)=Yε0h(x,y), where Y is the elastic modulus of the film, h(x,y) is the film thickness function, ε0 is the bulk strain in the film due to lattice mismatch which is assumed to be constant (not varying along z direction). However, as the island facets becomes steeper and island size becomes larger, the top part of the island feels less constraint from the substrate and becomes partially relaxed, i.e. the strain in the film


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(island) decreases along z direction.40-42 Such strain variation with z will inducing a nonlinear, higher-order dependence of surface stress on film thickness. In the simplest possible form, we assume the strain decreases linearly along z direction as ε(z)=ε0(1-τ0z),41,42 then the surface stress is expressed as:

σ ( x, y ) = σ 0  h( x, y ) − τ 0 h 2 ( x, y ) / 2  ,

(1)

where σ0=Yε0 is the bulk stress in the film due to lattice mismatch, h(x,y) is the film thickness depending on position, τ0 is nonlinear coefficient. Alternatively, the nonlinear elastic effect can simply be included by extending the linear stress-strain relationship [σ(x,y)=Yε0h(x,y)] to the second order as σ(x,y) ~ C1h(x,y) + C2h2(x,y), which is equivalent to Eq. (1), while the physical origin of the second-order term is shown to come from non-uniform strain distribution within the island. The first and second term in Eq. (1) represents the linear and nonlinear strain relaxation energy, respectively. For a faceted island formation, the strain relaxation energy can be calculated using the elastic Green's function method25,29-31, Eela = −

1 drdr ′χ ( r − r ′ ) f ( r ) f ( r ′ ) , 2∫

(2)

where χ is the elastic Green's function, r=(x,y) is the surface position, f(r) is the force monopole density on the island surface defined by fi(r)=∂iσ(r) where i=x, y. In the linear elastic regime, f(r) is a constant for a given facet; adding the nonlinear elastic effect, f(r) becomes thickness dependent and it is smaller at the apex region of the island. The island total energy on a surface [see Fig. 1 (upper inset)] can be obtained by integrating Eq. (2) and adding surface energy, which gives


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Etn = αV 2 / 3 − β V + τηV 4 / 3 ,

where α = ( sec θ − γ w / γ f

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(3)

) ( 6 cot θ )

2/3

, β = 9 tan θ / 2 and η = 64 / 3 tan 5 / 3 θ , γf and γw

are the surface energy density of the island facet and the wetting layer [Fig. 1 (upper inset)] , θ is the island facet angle. To obtain Eq. (3), we use γf3/c2 as energy unit, (γf/c)3 as volume unit, while c = σ 02 (1 −ν 2 ) / π Y is the scaled elastic energy density, ν is Poisson ratio, Y is the Young's modulus, V=L3tanθ/6 is the island volume, L is the island base length, τ is the scaled nonlinear coefficient.43,44 In the calculation below, we use γw/γf=1 for simplicity, i.e. we neglect surface energy anisotropy, which will not alter the general conclusion we obtain. The NLEF induces a strain term that increases with island volume in the power of 4/3 (keeping only the first order of τ term), which reduces the strain relaxation energy from the linear term and gives rise to a minimum of total energy at Vm, as shown in Fig. 1(b). Interestingly, this implies a thermodynamic self-limiting effect induced by nonlinear elastic term to hinder the island growth beyond Vm, since the island energy would increase with further growth, different from the previous works on interrupted coarsening due to kinetic limited growth rate, surface energy anisotropy, nonlinear stress field, or strain stabilization of 2D islands in linear regime.36-39 Consequently, beyond this “maximum” size, coherent island becomes more prone to injection of dislocations.1,17,33-35 Note that previous studies of the dislocation injection are based on comparison between the dislocation formation energy and island induced strain relaxation within the linear regime33,35; here we show that the dislocation formation may kick in sooner due to the nonlinear elastic strain relaxation within the island.


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Different from metal island, semiconductor strained island is usually faceted45, and the specific facet is often stabilized by strain, such as (105) facet of Ge.31 Below we will use Ge/Si(001) as a prototype system to do some quantitative analysis, and the facets involved including (105) for ‘pyramid’, (113) for ‘dome’ and (111) for ‘barn’ island6,17,34, where the facet angles (with respect to the substrate (001) surface) are 11.3, 25.2 and 54.7, respectively. Figure 2(a) shows the total energy Etn vs. the island size V at these three island facet angles. We can see that at small θ=11.3 (pyramid), Etn first increases with increasing V, reaches the maximum at critical size Vc (see inset), and then decreases as V further increases until it reaches the minimum at Vm and then increases again. At intermediate θ=25.2 (dome), the shape of Etn-V curve is similar with that of pyramid, while it crosses with pyramid energy line twice. This indicates that beyond certain size, pyramid would transform into dome. At even larger θ=54.7, Etn increases monotonically with island size and the critical size Vc going to infinity, which implies that large facet angle island (barn) would never form under such condition. The general dependence of Vc and Vm on θ is shown in Fig. 2(b) with the three typical θ values indicated. One can see that Vc increases with increasing θ. This is because when island size is small, surface energy plays the dominant role, formation of steeper island (large θ) forms with larger surface area for a given volume, leading to a larger critical size. This is consistent with the experimental observation that “pyramid” with shallow facet angle form first, followed later by “dome” with steeper facet angle or upon annealing.1,6,7 On the other hand, Vm decreases quickly as θ increases, indicating that steeper island has a smaller Vm. This is because steeper island has a larger height, maximizing the nonlinear effect. For large enough θ, Etn has no extremum, Vc goes to infinity.


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Vm being smaller for larger θ seemed to contradict our intuition, since in experiments dome is always larger than pyramid. However, in the experiment, the pyramid actually cannot reach its maximum size before transforming to dome or elongated to nanowire,25 while dome will reach its Vm without shape transition. This is also consistent with the experimental observation that dislocations usually form in the dome (presumably after it reaches Vm) but not in the pyramid without reaching Vm. Fig. 2(b) also shows that when NLEF is strong (large τ), the growth of too steep an island (large θ, e.g., barn island with (111) facet) is forbidden. This finding is consistent with the experimental observations that barn island cannot be found in pure Ge growth on Si substrate, while it appears in SiGe alloy island growth34 where NLEF should be smaller due to the larger island size and smaller bulk strain in the island comparing to the pure Ge island. Fig. 2(c) shows the dependence of Etn on τ. One sees that Vc does not vary much with τ because τ is a higher-order term, which only plays a role when V becomes large, so we can expect that the nonlinear elastic effect or non-uniform strain distribution would not affect the island nucleation behavior, such as, the nucleationless island formation.46 In contrast, Vm decreases sharply with increasing τ. This is further shown in Fig. 2(d) where Vc and Vm are directly plotted as a function of τ. The strong dependence of Vm on τ indicates that the self-limiting effect depends strongly on island facet angle, so does the dislocation formation within the island. In accordance with the experiment, we may perform some quantitative estimation of the nonlinear coefficient. As discussed above, the NLEF comes from the partial relaxation in the island along z direction. Usually the height of the ‘dome’ shape coherent island is about 10-30 nm6,34, if we assume the strain ε(z) relaxes to half of the bulk strain


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in the island at 30 nm, the nonlinear coefficient τ0 in Eq. (1) is 1/60 nm-1, the reduced nonlinear coefficient τ=τ0(γf/c). For Ge/Si(001) island, γf ≈ 6100 meV/nm2, c=270 meV/nm3,32 so we have τ≈0.38. Experiments showed that the size of Ge dome island ranges from 104 to 2×104 nm4; if we assume these are approximately their thermodynamic self-limited size, from Fig. 2(d) we estimate that the τ ranges from 0.35 to 0.42. Nevertheless, we also plot τ over a larger range in Fig. 2 just to reveal some general behavior. Now we consider the island grown inside a pit, as shown in Fig. 1 (lower inset). The solution of Eqns. (1) and (2) is similar as Eq. (3) and becomes Etnp = α ′V 2 / 3 − β ′V + τη ′V 4 / 3 ,

(4)

where α ′ = ( sec θ − γ wp sec ϕ / γ f )  6 / ( tan θ + tan ϕ ) 

2/3

,

β ′ = 9 ( tan θ + tan ϕ ) / 2 and

η ′ = 64 / 3 ( tan θ + tan ϕ ) , γwp are the surface energy density of the inclined pit surface, ϕ 5/3

is the pit inclination angle, and the island size in the pit is V=L3(tanθ+tanϕ)/6, where L is the length of the boundary formed between island facet and inclined pit surface, and we use the same reduced energy and volume units as for Eq. (3). Apparently, Eq. (4) reduces to the Eq. (3) for ϕ =0. The qualitative behavior becomes more complex due to the additional freedom, ϕ. Figure 3(a) shows the total energy of island in the pit Etnp as a function of island size V at different ϕ. One sees that for small ϕ (smaller than θ), Etnp also has a maximum at critical size Vc. The inset of Fig. 3(a) shows the enlarged plot for small island size V, and one can see that as ϕ increases, the critical size and energy barrier for island nucleation in the pit becomes smaller, same as the linear elastic analysis.30 For large ϕ, Etnp decreases from


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the beginning, i.e. the nucleation energy barrier reduces to zero. This is because when ϕ ≥ θ, α' becomes negative, the surface energy does not increase when island forms in the pit.

When ϕ < θ, i.e. α' >0, the behavior of Etnp is similar to Etn, only with different dependence on θ, ϕ and τ. Figure 3(b) shows the maximum island size Vm as a function of ϕ for different θ. For small θ, Vm deceases quickly with ϕ for small ϕ, and slows down as ϕ becomes larger; while for large θ, Vm does not change too much with ϕ. Especially, when θ is too large, at small ϕ, Vc goes to infinity, island cannot form; while at larger ϕ, Vc becomes finite and Vm appears, so island can form and grow. These results indicate that a pit can greatly enhance the nucleation of an island inside the pit, especially for large pit inclination angle. The total energy of the island in the pit shows similar dependence on the nonlinear coefficient τ as that of island on flat surface, i.e. Vm decreases with increasing τ; however, the maximum island size Vm becomes smaller because of the pitting effect, as shown in Fig. 3(c) in comparison with Fig. 2(c). Fig. 3(d) shows the dependence of Vm on τ for different ϕ; Vm decreases quickly at small τ, and slows down when τ becomes larger. For comparison, we also showed results without pit (ϕ=0), Vm becomes smaller as ϕ increases. Now we proceed to analyze how island may nucleate on pit-patterned substrates. First of all, at any pit inclination angle ϕ, island grown in the pit has a smaller nucleation energy barrier and critical size or no barrier compared with island grown on a flat surface, including at the rim of a pit. This means that island prefers to first nucleate and grow in the pit. In the experiments21,23, island can be found in all the pits; and for the pit with


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larger ϕ when additional islands form at the rim of pit, the islands in the pit are bigger, implying that they nucleate and grow first. Furthermore, our results suggest that on pit-patterned substrates, primary islands first nucleate in the pits and grow. When they reach the maximum size Vm whose growth slows down or stops, secondary islands may then nucleate at the rim. For pits with small inclination angle ϕ, islands grown in them have a very large maximum size, so that islands may never reach the maximum size and hence no secondary islands appear. Also, the pits may be too small, so that primary islands in the pits may grow to a large size to impede the nucleation of secondary islands at the rim next to them, due to the repulsive elastic interaction between islands.26,31 For pits with large ϕ, the maximum island size becomes smaller, islands in the pits can easily reach Vm, so that secondary islands are more likely to nucleate at the rim. Indeed, experiments found islands nucleate at the rim for larger ϕ, and the larger the ϕ is, the smaller the island size in the pit will be, and no island is expected at the rim if the pit size and/or inclination angle is very small. All these theoretical predictions are perfectly consistent with experimental observations.21,23 According to the discussion above, the maximum island size Vm is the key physical quantity in determining whether secondary islands can nucleate at the rim, and it depends on pit inclination angle, island facet angle and nonlinear strain relaxation. Thus, it is important to get a complete picture of how Vm varies with these parameters. Fig. 4 shows a 3D plot of Vm as a function of ϕ and θ. In general, for a given ϕ, Vm decreases as θ increases; for a given θ, Vm decreases as ϕ increases; however if θ is too large, island may not nucleate and grow at small ϕ (black region in Fig. 4). The nonlinear coefficient τ


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does not significantly affect the contour of Vm (ϕ, θ); however it affects the scale of Vm and the range of black region where the island does not form. In conclusion, we have developed a theoretical model to elaborate the NLEF on strained island growth on flat and pit-patterned substrates beyond the conventional linear elastic theory, which significantly advances our fundamental understanding of the strain island growth. We predict that the NLEF introduces a thermodynamic self-limiting effect on the growth of coherent islands, which depends on the island facet angle and pit inclination angle. In pit-patterned substrates, primary islands first nucleate and grow in the pit, then secondary islands may nucleate at the rim. Such a trend becomes stronger for larger and steeper pits. These findings explain several outstanding past and recent experimental observations.

AUTHOR INFORMATION Corresponding authors *E-mail: [email protected]

Notes The authors declare no competing financial interest.

ACKNOWLEDGMENTS The work at Xi'an Jiaotong University was supported by NSFC (Grant No. 11404252) and the Startup Funding from FIST, Xi'an Jiaotong University (H.H.). The work at University of Electronic Science and Technology of China was supported by the Startup Funding from UESTC(X. B.). The work at Utah was supported by DOE-BES (Grant No. DE-FG02-04ER46148) (F.L.).


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Figure Captions

Figure 1. Reduced island total energy in the linear (Etl) and nonlinear elastic regime (Etn) along with island surface energy (Es), linear strain relaxation energy (Els) and nonlinear strain relaxation energy (Ens) as a function of reduced island volume. Ec and Vc is the island nucleation barrier and critical size in linear elastic regime. γf, γw and γwp are the surface energy density of the island facet, the wetting layer on flat surface and inclined pit surface, respectively.

Figure 2. (a) and (c) Total energy of island vs. island volume V for different island facet angle θ (τ=0.3) and different nonlinear coefficient τ (θ=25.2), respectively. The inset in (a) shows the enlarged plot for small V. (b) and (d) Dependence of critical nucleation size Vc and maximum island size Vm on island faceted angle θ and nonlinear coefficient τ, respectively.

Figure 3. (a) and (c) Total energy of island grown in the pit vs. island size for different pit inclination angle ϕ and different nonlinear coefficient τ, respectively. The inset in (a) shows the enlarged plot for small V. (b) and (d) Dependence of the maximum island size Vm on pit inclination angle ϕ and nonlinear coefficientτ, respectively.

Figure 4. The maximum island size Vm of island grown in the pit as a function of island facet angle θ and pit inclination angle ϕ for (a) τ=0.4; (b) τ=0.6. The black region means Vm does not exists, i.e. the total energy increases monotonically with island size.


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Figure 1.


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Figure 2.


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Figure 3.


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Figure 4.


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(26) Shchukin, V. A.; Ledentsov, N. N.; Kop'ev, P. S.; Bimberg, D. Phys. Rev. Lett. 1995, 75, 2968. (27) Liu, P.; Lu, C.; Zhang, Y. W. Phys. Rev. B 2007, 76, 085336. (28) Xu, X.; Aqua, A-N.; Frisch, T. J. Phys.: Condens. Matter 2012, 24, 045002. (29) Katsaros, G.; Tersoff, J.; Stoffel, M.; Rastelli, A.; Acosta-Diaz, P.; Kar, G. S.; Costantini, G.; Schmidt, O. G.; Kern, K. Phys. Rev. Lett. 2008, 101, 096103. (30) Hu, H.; Gao, H. J.; Liu, F. Phys. Rev. Lett. 2008, 101, 216102. (31) Hu, H.; Gao, H. J.; Liu, F. Phys. Rev. Lett. 2012, 109, 106103. (32) Lu, G. H.; Liu, F. Phys. Rev. Lett. 2005, 94, 176103. (33) LeGoues, F. K.; Reuter, M. C.; Tersoff, J.; Hammar, M.; Tromp, R. M. Phys. Rev. Lett. 1994, 73, 300.

(34) Stoffel, M.; Rastelli, A.; Tersoff, J.; Merdzhanova, T.; Schmidt, O. G. Phys. Rev. B

2006, 74, 155326. (35) Johnson, H. T.; Freund, L. B. J. Appl. Phys. 1997, 81, 6081. (36) Jesson, D. E.; Chen, G.; Chen, K. M.; Pennycook, S. J. Phys. Rev. Lett. 1998, 80, 5156. (37) Aqua, J-N.; Frisch, T. Phys. Rev. B 2010, 82, 085322. (38) Pang, Y.; Huang, R. Phys. Rev. B 2006, 74, 075413. (39) Liu, F.; Li, A. H.; Lagally, M. G. Phys. Rev. Lett. 2001, 87, 126103. (40) Steinfort, A. J.; Scholte, P. M. L. O.; Ettema, A.; Tuinstra, F.; Nielsen, M.; Landemark, E.; Smilgies, D.-M.; Feidenhans'l, R.; Falkenberg, G.; Seehofer, L.; Johnson, R. L. Phys. Rev. Lett. 1996, 77, 2009. (41) Liu, F.; Huang, M. H.; Rugheimer, P. P.; Savage, D. E.; Lagally, M. G. Phys. Rev. Lett. 2002, 89, 136101.


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Nano Letters

(42) Zinovyev, V. A.; Vastola, G.; Montalenti, F.; Miglio, Leo. Surf. Sci. 2006, 600, 4777. (43) It can be shown τ is proportional to τ0 but independent of θ, and

τ = 2(4 ln 2 − 1)τ 0 / 3 in 2D but its analytical form in 3D is unknown. (44) Note that our expression of the linear term has a small difference from ref. 25, because we use the exact volume expression instead of V=L3tanθ/8. Also the units for energy and volume we use are not the critical size and energy barrier in the linear regime, because we need to examine the dependence on θ, so the reduced units cannot depend on θ.

(45) Liu, F. Phys. Rev. Lett. 2002, 89, 246105. (46) Sutter, P.; Lagally, M. G. Phys. Rev. Lett. 2000, 84, 4637.


Nano Letters

3

Linear Elastic Regime

Strain field

Substrate

Total energy

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0

Etn

Nucleation barrier

Etl

-3

Nonlinear Elastic Regime

Self-limiting size

Strain field -6 0.01

Substrate


0.1

1

10

Island volume

100

Thermodynamic Self-Limiting Growth of Heteroepitaxial Islands

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