Thermodynamic stability analysis of isometric and elongated epitaxial ${\text{Ce}}_{1\ensuremath{-}x}{\text{Gd}}_{x}{\text{O}}_{2\ensuremath{-}y}$ nanostructures on perovskite substrates

Thermodynamic stability analysis of isometric and elongated epitaxial ${Ce}_{1 - x} {Gd}_{x} O_{2 - y}$ nanostructures on perovskite substrates

Marta Gibert, Alberto García, Teresa Puig, and Xavier Obradors

Phys. Rev. B 82, 165415 – Published 7 October 2010

Abstract

The identification of the equilibrium shape of strain-induced nanostructures plays a key role for the nanoengineering of many materials. In this paper, thermodynamic analyses are used to model the observed different shapes of epitaxially grown ${Ce}_{1 - x} {Gd}_{x} O_{2 - y}$ (CGO) nanoislands on oxide perovskite single-crystals substrates. We investigate the role of modified biaxial strain arising from islands with different crystallographic orientations. We show that the in-plane lateral aspect ratio (isomorphic, anisotropic) of the interfacial nanoislands can be tuned through the strains with the thermodynamic results reported being consistent with the shapes and sizes experimentally observed. The equilibrium shape of isotropic-strained (001)CGO nanoislands corresponds, within the investigated sizes, to square-based nanopyramids and they have a stable size. Instead, the strain anisotropy of (011)CGO nanoislands is proved to be essential to explain their high shape anisotropy, which increases continuously with island coarsening as a mechanism to minimize their energy density. This model sheds light on the key parameters involved in a very general approach for the generation of epitaxial nanowires.

Received 4 August 2010

DOI:https://doi.org/10.1103/PhysRevB.82.165415

Authors & Affiliations

Marta Gibert, Alberto García, Teresa Puig, and Xavier Obradors^*

Institut de Ciència de Materials de Barcelona (ICMAB)-CSIC, 08193 Bellaterra, Catalonia, Spain

^*xavier.obradors@icmab.es

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Vol. 82, Iss. 16 — 15 October 2010

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Figure 1
(Color online) Strain-induced CGO nanostructures on LAO single-crystal substrates. (a) Isomorphic square-based (001)CGO nanodots. Inset: enlarged view of the same nanostructures. (b) Anisotropic (011)CGO nanowires aligned along substrate soft axes {100}. Schematic representations of the orientation of (c) (001)CGO and (d) (011)CGO nanostructures on (001)LAO substrates. (e) Measured lateral aspect ratio $c$ as function of effective diameter $D$ for both (001) dots (triangles) and (011) wires (dots). Notice the stability in shape and size of nanodots as compared to nanowires, which continuously elongate as their in-plane size increases.Reuse & Permissions
Figure 2
(Color online) Schematic representation of the assumed three-dimensional island’s shape, showing the width $a$ , the length $b$ , the height $h$ , and the contact angles $δ$ and $θ$ .Reuse & Permissions
Figure 3
(Color online) Energy of (001)CGO nanoislands on LAO substrate as function of its lateral aspect ratio $c$ . (a) Total energy (red solid line), relaxation energy (blue dashed line), and surface energy (green dot line) vs lateral aspect ratio $c$ for a typical nanoisland of size $D = 20 nm$ and $h = 6.5 nm$ . (b) $E (c)$ calculations for different island sizes, $D = 10$ (green, lower line), 20 (red), 30 (blue), 40 (orange), 50 (purple), and 60 (cyan, upper line) nm. Parameters $α = 10^{7} J m^{- 3}$ , $β = 1$ , and $A$ within the range 0.5 and 3 nm were considered. The energy minima at $c_{e q} = 1$ indicates that the equilibrium shape of (001)CGO nanoislands corresponds to a square-based nanopyramid for all the investigated sizes.Reuse & Permissions
Figure 4
(Color online) Phase diagram showing the thermodynamically derived in-plane anisotropic shape of (011)CGO nanoislands on LAO substrates as function of $α$ and the anisotropic strain parameter $β$ . Different behaviors are distinguished depending on the parameter region.Reuse & Permissions
Figure 5
(Color online) Interfacial (011)CGO nanoislands on LAO substrate. (a) $E_{t o t a l} (c)$ (red solid line), $E_{s u r f} (c)$ (green dot line), and $E_{r e l a x} (c)$ (blue dashed line) corresponding to an island of $D_{1} = 40 nm$ and $h = 3.3 nm$ . The energy minima at $c_{e q} > 1$ indicates that the equilibrium shape is a rectangular island. (b) $E (c)$ calculations corresponding to island’s sizes $D_{1}$ (blue solid line) and $2 D_{1}$ (orange dashed line). The arrows indicate the equilibrium lateral aspect ratio $c_{e q}$ for each size. The shift of the energy minima as island size increases shows that it elongates when growing. (c) Equilibrium anisotropic shape $c_{e q}$ for different effective diameters; good agreement between dimensions measured by TEM (open circles) and thermodynamically calculated data (squares) is appreciated. Inset: fitting parameters $A$ considered in the main plot $c_{e q} (D)$ for each island height $h$ corresponding to a given size. Parameters $α = 10^{8} J m^{- 3}$ , $β = 25$ , and $k^{'} = 1.1$ were used for the calculations.Reuse & Permissions
Figure 6
(Color online) Energy density $E / V$ as function of the effective diameter $D$ of CGO nanoislands on LAO substrates. (a) $E / V (D)$ for an isomorphic (001) nanodot. Inset: $E / V (D)$ (red solid line), $E_{s u r f} / V (D)$ (green dot line), and $E_{r e l a x} / V (D)$ (blue dashed line). An optimum size $D_{o p t} = 20 nm$ is determined considering $h = 7 nm$ and $β = 1$ , and adjusting $α \sim 10^{8} J m^{- 3}$ and the fitting parameter $A = 3.5 nm$ . (b) $E / V (D)$ for a (011)CGO nanoisland. Parameters $α = 10^{8} J m^{- 1}$ , $β = 25$ , and $k^{'} = 1.1$ were used in the computation. Notice that there exists an optimal island size for (001) nanoislands, whereas coarsening is promoted between (011) ones.Reuse & Permissions

Physical Review B

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Thermodynamic stability analysis of isometric and elongated epitaxial ${Ce}_{1 - x} {Gd}_{x} O_{2 - y}$ nanostructures on perovskite substrates

Marta Gibert, Alberto García, Teresa Puig, and Xavier Obradors

Phys. Rev. B 82, 165415 – Published 7 October 2010

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Thermodynamic stability analysis of isometric and elongated epitaxial Ce1−xGdxO2−y nanostructures on perovskite substrates

Marta Gibert, Alberto García, Teresa Puig, and Xavier Obradors

Phys. Rev. B 82, 165415 – Published 7 October 2010

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Thermodynamic stability analysis of isometric and elongated epitaxial ${Ce}_{1 - x} {Gd}_{x} O_{2 - y}$ nanostructures on perovskite substrates