Metastability in 2D Self-Assembling Systems

We show that 2D self-assembled domains can remain trapped in a large variety of long-lived and metastable shapes that arise from an interplay of crystalline anisotropy and relaxation of elastic strain. On commonly used cubic (111) substrates, these shapes include extended or stacked structures made up of triangular domains connected at their corners, compact shapes with both convex and concave curvatures and others with narrow and elongated arms. We show that all of these distinct experimentally observed shapes can be explained within a unified framework and present a phase diagram that systematically classifies the metastable shapes as a function of their size.

Figure 1

(a) LEEM images of ($7\times 7$) reconstructed domains on a ($1\times 1$) reconstructed Si(111) surface. The domain boundaries are oriented along $⟨110⟩$. The areas of the domain marked 1, 2, $2^{\prime }$, 3, 4, $4^{\prime }$ are 0.76, 0.78, 1.33, 0.72, 1.31, and $1.32\mu {\mathrm{m}}^2$, respectively. (b) STM image of Co islands on Pt(111), with epitaxial mismatch strain of 9% [7]. The three arms of Co islands are 3–5 nm wide and up to 250 nm long. They run perpendicular to the close-packed $⟨110⟩$ orientations. Reproduced with the permission of the authors [7].

Figure 2

Equilibrium shapes of domains on (111) surfaces given for three different values of the anisotropy parameter, $\alpha$. The domains marked with the * are in metastable equilibrium. The inset in (b) shows the plot of $\beta (\theta )$ with three minima and three maxima labeled A–C and D–F, respectively, and six intermediate orientations labeled 1–6. These orientations are also shown in the equilibrium shapes in (b).

Figure 3

Energy per unit area for: (a) domain shapes on a cubic (111) surface given in Fig. 2, and (b) square and rectangular domains on a (001) surface [4]. This quantity attains minimum for the compact domain of size $0.2\mu {\mathrm{m}}^2$ on the (111) surfaces, while elongated nanowire shape has smaller minimum energy per unit area on the (001) surface.

Figure 4

Energy landscapes for the growth of notches plotted as a function of their location ($s_1$) and lengths ($s_2$): (a) One notch on a domain with area $0.6\mu {\mathrm{m}}^2$, and (b) Two notches on a domain with area $1.0\mu {\mathrm{m}}^2$. Insets show the evolution of domain shapes and the corresponding energies along saddle paths. Material parameters used in the calculations, ${\beta }_0=15.48\mathrm{meV}/\mathrm{nm}$, $C_0=2.853\mathrm{meV}/\mathrm{nm}$ correspond to the ($7\times 7$) domains on Si(111).

Figure 5

Morphological phase diagrams for strained domains on (111) surface as a function of normalized area $A/A_o$ and anisotropy factor $\alpha$. Part (a) shows optimum connected-domain shapes while part (b) shows the optimum shapes of an isolated compact domain. While energy barriers are encountered in crossing the phase boundaries in (a), the phase boundaries in (b) involve continuous transitions without any barriers. The energy difference between the different shapes with 3 and 4 connected domains is very small ($<5\%$). Points marked 1 and 3 correspond to the Si ($7\times 7$) domains of nearly identical size in Fig. 1a. Note that although the total energy of domain 3 is lower, the domain 1 is trapped in a metastable state. The anisotropy parameter for this material system is $\alpha =0.28$ [11].