Stress-driven island growth on top of nanowires

We model the coherent deposition of a mismatched material on the top facet of a nanowire and investigate the possible formation of a cylindrical island that is narrower than the nanowire stem. We calculate the elastic relaxation and the total energy of the system and determine the optimal shape of the deposit as a function of misfit, nanowire radius, and deposit thickness. If the values of any two of these parameters are set, then there is a critical value of the third one above which the formation of a genuine island that is narrower than the stem is favored compared to that of a disk of equal volume covering entirely the top facet. These critical values are easily accessible in current nanowire systems, and we predict that islanding can lead to a large reduction of the total energy. We discuss the similarities and differences between the present effect and the standard Volmer-Weber and Stranski-Krastanow growth modes. For semiconductor nanowires, islanding is likely to occur primarily in the case of catalyst-free growth. We argue that it might already have been observed in heterostructures of nanowires of group-III nitrides.

Figure 1

Two scenarios for the spontaneous formation of a quantum dot on top of a NW. (a) Continuous growth of radially inhomogeneous alloy monolayers. (b) Island growth followed by conformal shell embedding.

Figure 2

Schematics of islands of equal volumes covering the top facet of the nanowire either (a) partly or (b) fully (disk); (c) equivalent volume as part of a 2D layer grown on a bulk substrate.

Figure 3

Variations of the ratio of the elastic energy stored in the system made of a misfitting island coherently deposited on a NW stem [Fig. 2a] to that stored in a volume equal to the island, cut in a 2D layer [Fig. 2c], with the layer/substrate and island/NW misfits being equal. Variations are given as a function of island height $h$ for various values of the island radius $r$, with $h$ and $r$ in units of NW radius $R_0$, and $\nu =0.33$. Each symbol represents a numerical calculation made for a given island geometry. The curves are fits of these data with functions ${\phi }_{\nu }$ given by Eqs. (4) and (5). Since $w_e$ gets smaller and varies more slowly at high $h$, the graph is divided in two halves for the sake of visibility: starting from $h=0.5$, the horizontal scale is contracted by a factor of 7.2 and the vertical scale is expanded by a factor of 10 (right axis).

Figure 4

Map of the variations as a function of NW radius and nominal deposit thickness of the ratio $W_{\mathrm{opt}}/W_{\mathrm{disk}}$ of the minimum energy of the system (at given deposit volume) to its energy in the disk configuration. The nominal deposit thickness is expressed in monolayers of thickness 0.27 nm. The misfit ${\epsilon }_0$ of the deposit with respect to the NW is (a) 2$\%$ or (b) 3$\%$. The gray zones at left and bottom are where the disk is favored (ratio equal to 1), whereas island formation is preferred in the colored areas. Material parameters: $\nu =0.33$, $E=3.24\times {10}^{11}\text{Pa}$, ${\gamma }_{SW}=1.8\text{J}/{\text{m}}^2$.

Figure 5

Same as Fig. 4 for ${\epsilon }_0=4\%$. The arrows illustrate the existence of a deposit-thickness-dependent critical NW radius and a NW-radius-dependent critical deposit thickness.

Figure 6

Variation with island radius, at fixed island volume $V$ and for a given NW radius $R_{\mathit{NW}}=25\text{nm}$, of the elastic (red curves), sidewall (blue curves), and total (green curves) energies. Full and dashed curves correspond, respectively, to deposit thicknesses of 5 and 0.2 MLs (at full NW radius). The energies are normalized to the total energy of the disk with the same volume. Material parameters as in Figs. 4 and 5; ${\epsilon }_0=4\%$.

Figure 7

Map of the variations as a function of NW radius and deposit thickness of the radius of the optimal island. The radius is expressed in units of NW radius. Material parameters as in Fig. 5.

Figure 8

Same as Fig. 5 with nonzero differences of surface energies between the top facets of the island and NW stem: (a) $\Delta \gamma =-0.18\text{J}/{\text{m}}^2$, (b) $\Delta \gamma =0.18\text{J}/{\text{m}}^2$. The dashed line in (b) marks the disk/island transition when elastic energy is ignored (see text).