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DOI: 10.1039/C9NA00741E
(Paper)
Nanoscale Adv., 2020, Advance Article

Mourad Mezaguer,
Nedjma Ouahioune and
Jean-Noël Aqua*

Sorbonne Université, CNRS, Institut des Nanosciences de Paris, INSP, UMR 7588, 4 Place Jussieu, 75005 Paris, France. E-mail: aqua@insp.jussieu.fr

Received
23rd November 2019
, Accepted 11th January 2020

First published on 13th January 2020

We investigate the influence of strain-sharing and finite-size effects on the morphological instability of hetero-epitaxial nanomembranes made of a thin film on a thin freestanding substrate. We show that long-range elastic interactions enforce a strong dependence of the surface dynamics on geometry. The instability time-scale τ is found to diverge as (e/H)^{−α} with α = 4 (respectively 8) in thin (resp. thick) membranes, where e (resp. H) is the substrate (resp. nanomembrane) thickness, revealing a huge inhibition of the dynamics as strain sharing decreases the level of strain on the surface. Conversely, τ vanishes as H^{2} in thin nano-membranes, revealing a counter-intuitive strong acceleration of the instability in thin nanomembranes. Similarly, the instability length-scale displays a power-law dependence as (e/H)^{−β}, with β = α/4 in both the thin and thick membrane limits. These results pave the way not only for experimental investigation, but also, for the dynamical control of the inescapable morphological evolution in epitaxial systems.

We consider in the following a hetero-epitaxial nano-membrane where a thin crystalline film of thickness h is coherently deposited on a thin substrate of thickness e that is supposed to be freestanding and flat. The lattice mismatch between the film and substrate generates strain, and the long-range elastic field penetrates throughout the system, building an explicit dependence on geometry. First, strain sharing occurs between the film and substrate and is quantified by the ratio e/H (with H the system thickness H = e + h). Second, any modulation of the surface with a lateral extension λ produces a field that extends also down to λ in the film and substrate, leading to a dependence on H (or more precisely on H/λ).^{24} These two strain-sharing and finite-size effects introduce a new way to tune strain at will thanks to geometry. In addition, it is known that the strain thus produced may cause the morphological evolution of the surface when surface diffusion is active. This is basically described by the Asaro–Tiller–Grinfeld (ATG) instability^{25,26} that is especially at work in SiGe systems at low strain^{27} (as opposed to the nucleation occurring at higher strain^{28}). We therefore revisit this instability to investigate the influence of finite-size effects and strain sharing on the dynamics of the growth of a film deposited on a nanomembrane substrate. We thence focus on the growth dynamics of the film, and not on equilibrium effects such as the ones for example that rationalize ordering of quantum dots on nanomembranes thanks to energetic considerations, see e.g. ref. 29–31.

In the following, we compute first the strain field generated in a hetero-epitaxial nanomembrane with free boundary conditions, corresponding to ultra-high vacuum conditions. We compute strain both in the flat film geometry and for a modulation with small slopes. This solution at linear order allows us to compute analytically the surface dynamics due to surface diffusion for a single harmonic. By Fourier decomposition, we then compute the surface evolution during annealing. We show that the dynamics is strongly affected by both finite-size and strain-sharing effects, with a possible dynamical inhibition or conversely strong acceleration of the morphological instability. The characteristic time and length scales are then shown to behave algebraically as a function of e/H and H.

(1) |

∇·σ = 0, | (2) |

σ·n_{z}(z = 0) = 0
| (3a) |

σ·n(z = H) = 0 | (3b) |

u(z = e^{−}) = u(z = e^{+}),
| (4a) |

σ·n_{z}(z = e^{−}) = σ·n_{z}(z = e^{+}).
| (4b) |

Fig. 1 Geometry of a nanomembrane with a film coherently deposited on a thin substrate of finite thickness. |

The computation of the stress field may be done as a power-law expansion in the small-slope approximation where |∇H| ≪ 1. Its results depend crucially on the level of stress in the zeroth-order flat-film geometry^{34} where H(r) = H when the system is invariant by translation or rotation in the (x, y) plane. Hence, all the measurable properties such as forces, stress tensor and displacement gradients are independent of x and y, but not necessarily the displacement vector defined only up to an arbitrary reference state. In this geometry, the displacement vector u_{0} satisfies

(5) |

Given the invariance of σ on x and y, Navier eqn (2) projected on the x direction leads to the fact that σ_{xz}, and thence e_{xz} and ∂u_{0,x}/∂x, are constant both in the film and substrate. Similarly, the projection of (2) on the z-direction leads to a constant σ_{zz} and ∂u_{z}/∂z in the film and in the substrate, and to the same conclusion for ∂u_{0,z}/∂x and ∂u_{0,z}/∂y (after differentiation of σ_{zz} with respect to x). The general solution for the mechanical equilibrium accounting for the flat geometry is then

(6) |

(7) |

(8) |

In the limit of a semi-infinite substrate, eqn (8) leads to the known result that vanishes in the substrate and displays the Poisson's dilatation in the film. In the opposite limit e → 0, one finds the symmetric case where the film is fully relaxed while the substrate displays the Poisson's dilatation in the opposite direction. In between, the solution (8) quantifies the strain shared between the film and the substrate. Finally, the elastic energy density associated with (8) is in the film

(9) |

We now turn to the case where the film is corrugated and displays small slopes. Writing H(r) = H + h_{1}(r) with H = 〈H(r)〉, one may find the solution for the displacement vector as an expansion u = u_{0} + u_{1} +…, supposing that h_{1} (in fact |∇h_{1}|) is a small parameter. At equilibrium, u_{1} may be conveniently found in Fourier space in the x and y directions, with the result given in Table 1.

We find there are six unknown C^{α}_{i}s both in the film and substrate, and, contrarily to the semi-infinite case, both e^{kz} and e^{−kz} terms, with the wavevector k and k = |k|. The boundary conditions at the interface (4a) and (4b) give C^{f}_{i} = C^{s}_{i} ≡ C_{i} (independent of the film or substrate) for i = 1…6, that lead to u^{f}_{1} = u^{s}_{1}, an identity resulting from the hypothesis of an identical film and substrate elastic constants.† The boundary condition (3a) gives C_{2} = ik_{x}C_{5}, C_{4} = ik_{y}C_{5} and C_{6} = iν(k_{x}C_{1} + k_{y}C_{3})/(1 − ν). Eventually, the surface boundary condition (3b) gives

(11) |

(12) |

Given this solution for the C_{i}s and thence for u (given explicitly in Appendix), one can compute the elastic energy density on the film surface at z = H(r), that reads with given in (9) and

(13) |

(14) |

In the limit of a semi-infinite substrate (where e ≫ h and kH ≫ 1), one finds as expected.^{35} Otherwise describes the influence of strain-sharing and of finite-size effects on elasticity.

(15) |

We plot in Fig. 2, the resulting growth rate as a function of k and e for different membrane thicknesses H. It is first noted that σ can be either strongly increased or lowered depending on finite-size effects (ruled by H) and strain sharing (ruled by e/H). To quantify this, we compute the maximum of σ for a given e and H, that occurs at (k_{max}, σ_{max}), see Fig. 3. We take as a reference, the infinite-substrate limit σ_{∞}(k) = k^{3} − k^{4} for which σ^{∞}_{max} = 27/256 ≃ 0.105 and k^{∞}_{max} = 3/4 (this limit occurs when both H ≫ 1 and e/H ≃ 1, i.e. e ≫ h). This limit is already nearly achieved when H = 10 and when strain sharing vanishes (e/H ≃ 1). By decreasing the membrane thickness, one finds a ten-fold increase in σ_{max} for H = 1 without strain sharing (σ_{max} ≃ 1.15 when e/H ≃ 1) and a 10^{3}-fold increase for H = 0.1 (σ_{max} ≃ 101 when e/H ≃ 1). Hence, for a given e/H, the maximum growth rate increases with H, showing the a priori counter-intuitive influence of finite-size effects that enforce a faster relaxation for a thinner membrane. Conversely, for a given H, the growth rate significantly decreases when strain sharing occurs (i.e. when e/H decreases from 1). For H = 10, one finds respectively σ_{max} = 0.105, 9.5 × 10^{−4} and 1.0 × 10^{−6} for e/H = 1, 1/2 and 1/10. Similarly, for H = 0.1, one finds respectively σ_{max} = 101, 6.27 and 1.0 × 10^{−2} for e/H = 1, 1/2 and 1/10. Therefore, the stronger strain-sharing is (i.e. the lower e/H is), the slower the instability occurs, as the less strained the system is.

The second conclusion regarding σ(k, e, H) is the variation of the maximum wavelength k_{max} as a function of finite-size and strain-sharing effects. We find, see Fig. 2, that for a given H, k_{max} decreases when strain-sharing increases (i.e. when e/H decreases) while, for a given e/H, k_{max} increases when finite-size effects increase (i.e. when H decreases). Numerically, we find for H = 10, respectively k_{max} = 0.75, 0.19 and 0.032 for e/H = 1, 1/2 and 1/10, while for H = 0.1, k_{max} = 3.18, 1.58 and 0.32 for e/H = 1, 1/2 and 1/10. The decreases of k_{max} with strain-sharing corroborate the fact that the film is globally less strained in this case. Conversely, the increase in k_{max} with finite-size effects is consistent with the increase in σ, signaling the increase in the surface strain in this case. Globally, even if the variation of k_{max} with e/H and H is quantitatively less pronounced than for σ_{max}, it is nonetheless significant and leads to variations that are expected to be important in experimental systems.

We note that for given strain-sharing and finite-size effects, the growth rate eqn (15) always displays a positive maximum so that the morphological instability should always occur (we neglect here the influence of wetting effects that can lead to the existence of a critical thickness,^{35} in order to focus solely on the influence of strain-sharing and finite-size effects). Indeed, for given e/H and H, we find at low-k ‡ while σ(k, e, H) ≈ −k^{4} at large k.§ Another interesting limit is the thin-membrane limit (when H ≪ 1), where with = k/k_{t} and k_{t} = e/H^{3/2}. When both e and H are of order ε, k_{max} diverges as while σ_{max} behaves as 1/ε^{2}.

The resulting typical evolution on top of a membrane is shown in Fig. 4. We characterize the surface geometry with the length-scale λ that can be related to the average wave-vector

(16) |

We plot in Fig. 5 and 6, the resulting characteristic scales for H = 100, 1 and 0.1. It is noteworthy that the typical time scale τ displays huge variations as a function of strain sharing and finite-size effects. For a given H, it shows a 10^{4} (respectively 10^{2}) increase when e/H decreases from 1 to 0.4 for H = 100 (resp. 0.1), describing a strong slowdown of the instability evolution. This sensitivity is naturally related to the decrease in the global strain with strain-sharing, as described above for the growth rate σ. But we also find that τ decreases strongly when H decreases for a given e/H: we find e.g. a decrease from 30 down to 0.012 in between H = 100 and 0.1 for e/H = 1. This reveals the counter-intuitive result of these finite-effects, coherently related to their influence on the growth rate σ: a thinner membrane leads to a much faster surface dynamics, and the acceleration of the instability. Similarly, the typical wave-length of the instability is also ruled by strain-sharing and finite-size effects, but with a lower amplitude, see Fig. 6. The increase in λ when e/H decreases is again related to the decrease in the global strain but with a factor at most around 3 to 4, while its decrease when H decreases is also related to the counter-intuitive finite-size effects. In addition, we find that the resulting length and time scales λ and τ are very well approximated by λ_{max} and τ_{max}, showing that the fastest growing mode is quickly driving the surface dynamics. It is nonetheless not a perfect approximation, especially for τ and at low thickness H where the maximum of σ(k) is less sharp.

To get some analytical insights on these evolutions, we find that the previous results can be well approximated in some limits, see Fig. 5 and 6. When finite-size effects vanish, i.e. for H ≫ 1, one finds

(17) |

(18a) |

(18b) |

These approximations are plotted in Fig. 5 and 6 and do indeed perfectly capture the numerical estimate of τ_{max} and λ_{max}, and approximate well the global time and length scales τ and λ. In the other limit of strong finite-size effects, i.e. for H ≪ 1, we find the Laurent series

(19) |

(20a) |

(20b) |

(21) |

D_{x} = i2k_{x}{ksinh(kz) × … × [((2ν − 1)H − z)sinh(kH) + kHzcosh(kH)] − cosh(kz)[(k^{2}Hz − 2(ν − 1))sinh(kH) + 2(ν − 1)kHcosh(kH)]}
| (22) |

D_{z} = −k{sinh(kH) × … × [sinh(kz)(kz − k^{2}Hz − 2(ν − 1)) − 2cosh(kz)(kz − 2(ν − 1)kH)] + 2kHcosh(kH)[(2ν − 1)sinh(kz) + kzcosh(kz)]}
| (23) |

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## Footnotes |

† This was also proven in the semi-infinite substrate case. |

‡ Note that the k → 0 and H → ∞ limits do not commute. |

§ Subsequently, there exists k_{+} such as σ < 0 for k > k_{+}, and the instability will occur only if no lateral finite-size effect occurs, i.e. only if L > 2π/k_{+}. |

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