# Can Nanowires Coalesce?

## Abstract

**:**

## 1. Introduction

## 2. Model

## 3. Results and Discussion

_{x}/Si(111) substrates [36], and is explained by the enhanced shadowing of denser NWs and lower material flux per NW contributing to the radial growth [28,29,37]. It does not contradict the fact that merging of NWs occurs earlier in denser NW ensembles, simply because the late stage of growth corresponding to the full shadowing of the substrate surface starts earlier than in sparse ensembles (corresponding to shorter times ${t}_{*}$ in Equation (1)). This effect will be considered in detail in what follows.

^{11}cm

^{−2}[6,7,11,32,40], which is why the coalescence process starts quite early. The typical base radius of the initial 3D islands equals 5 nm [7,32,37]. Desorption of Ga adatoms from the NW sidewalls should be present in high-temperature MBE growth of GaN NWs at around 800 °C [32]. According to Equation (22), in the initial stage of growth of separated GaN NWs, corresponding to $\gamma a(R-{R}_{1})/{\mathsf{\Lambda}}_{inc}\ll 1$ and $c{R}^{2}N\ll 1$, the NW length scales as a power law of its radius:

^{10}, 5 × 10

^{10}and 10

^{11}cm

^{−2}, corresponding to the NW separations $P=$ 100, 45 and 32 nm. The curves are obtained from Equation (22) at $a=1$ in the early stage and from numerical solution of Equations (2) and (4) after the flux saturation. These curves show the importance of the NW surface density (determined in the nucleation stage of 3D islands in the case of SI GaN NWs) in the kinetics of the coalescence process. The low surface density NWs ($N=$ 10

^{10}cm

^{−2}, $P=$ 100 nm) remain separated even at a large $L$ of 2000 nm, although they will finally coalesce due to $a=1$. The higher surface density NWs start to merge in the early stage of growth. The highest surface density NWs ($N=$ 10

^{11}cm

^{−2}, $P=$ 32 nm) are almost fully coalesced at $L=$ 2000 nm.

## 4. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Stages of the growth process resulting in (

**a**) isolated NWs, (

**b**) partly merged NWs, and (

**c**) fully coalesced NWs with $\theta =1$. (

**d**) Mosaic structure of merged SI GaN NWs [7] in stage (

**b**). A similar picture of the coalescence process applies to 3D islands in SAE, where the islands emerge with a small size and a given density (determined in the nucleation step) but then merge together to fill a growth template.

**Figure 2.**Normalized perimeters per unit surface area $u\text{}$and per surface area of the NW tops $u/\theta $ versus $\theta $ (the solid lines). The dashed lines show the same functions for isolated NWs.

**Figure 3.**Asymptotic coverage ${\theta}_{\infty}$ versus $a$ at three different $b$ shown in the legend (the solid lines), obtained from Equation (10). The dashed lines show the approximations given by Equation (9) for isolated NWs, which underestimate the coverage for any $a$ at $b>0$. At $b=0$, both models give the same result for ${\theta}_{\infty}=1/a$. The full coalescence occurs only at $a=1$. The asymptotic coverage at a fixed $a>1$ decreases with $b$. The shaded zone corresponds to partly merged NWs at ${\theta}_{\infty}>{\theta}_{m}=0.39$.

**Figure 4.**Maximum coverage of a substrate surface with NWs that start from a fixed radius of 30 nm and have a fixed surface density corresponding to an average separation of 500 nm, at different $a$ shown in the legend. NWs will fully coalesce only at $a=1$, as in Figure 3. Other NWs will only partly merge.

**Figure 5.**Maximum asymptotic coverage as a function of the droplet contact angle for VLS NWs grown by VPE and MBE at $\alpha =$ 45° and 30°. VPE-grown NWs will only partly merge but never coalesce into continuous film. MBE-grown NWs will fully coalesce at small enough droplet contact angles $\beta \le {90}^{\xb0}-\alpha $.

**Figure 6.**Evolution of the coverage with the length of catalyst-free NWs for three different NW surface densities corresponding to the average separations $P$ shown in the legend. At a low surface density of 10

^{10}cm

^{−2}($P=$ 100 nm), NWs do not even start to merge at an average length of 2000 nm, while the higher surface density NWs with $N=$ 5 × 10

^{10}and 10

^{11}cm

^{−2}merge early. At the highest surface density of 10

^{11}cm

^{−2}, the NW film almost reaches continuity at $L=$ 2000 nm.

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**MDPI and ACS Style**

Dubrovskii, V.G.
Can Nanowires Coalesce? *Nanomaterials* **2023**, *13*, 2768.
https://doi.org/10.3390/nano13202768

**AMA Style**

Dubrovskii VG.
Can Nanowires Coalesce? *Nanomaterials*. 2023; 13(20):2768.
https://doi.org/10.3390/nano13202768

**Chicago/Turabian Style**

Dubrovskii, Vladimir G.
2023. "Can Nanowires Coalesce?" *Nanomaterials* 13, no. 20: 2768.
https://doi.org/10.3390/nano13202768