Diameter dependence of the growth velocity of silicon nanowires synthesized via the vapor-liquid-solid mechanism

We present a model for the radius dependence of the growth velocity of Si nanowires synthesized via the vapor-liquid-solid mechanism. By considering the interplay of the Si incorporation and crystallization rate at steady state conditions we show that the radius dependence of the growth velocity in general depends on the derivatives of the incorporation and crystallization velocity with respect to the supersaturation. Taking this into account, the apparently contradictory experimental observations regarding the radius dependence of the growth velocity can be reconciled and explained consistently.

Figure 1

(a) Schematic of the VLS mechanism: (i) incorporation, (ii) diffusion, (iii) crystallization. (b) Chemical potentials (CPs): ${\mu }_0^s=\mathrm{CP}$ of the Si substrate, ${\mu }^v=\mathrm{CP}$ of the Si vapor, ${\mu }^s=\mathrm{CP}$ of the Si nanowire (NW), ${\mu }^{vl}=\mathrm{CP}$ difference between vapor and liquid, ${\mu }^{ls}=\mathrm{CP}$ difference between liquid and Si NW, $\xi =\mathrm{CP}$ difference between vapor and Si NW.

Figure 2

Diameter dependence of the growth velocity; all experiments performed with $\mathrm{Si}{\mathrm{Cl}}_4$ as precursor. (a) Growth velocity $v$ as a function of the wire diameter $d$; $\mathrm{Si}{\mathrm{Cl}}_4$ pressure increases from (1) to (4); after Ref. 7. (b) $v$ as a function of $d$; $1130°\mathrm{C}$; $\mathrm{Si}{\mathrm{Cl}}_4∕{\mathrm{H}}_2=6.7\%$; after Ref. 7. (c) $v$ as a function of $d$; temperature $1000–1100°\mathrm{C}$; (1) $\mathrm{Si}{\mathrm{Cl}}_4:{\mathrm{H}}_2=0.9\%$, (2) 0.95%; after Ref. 9. (d) $v$ as a function of $d$; temperature (1) 1027, (2) 1047, (3) 1067, (4) 1087, and (5) $1107°\mathrm{C}$; after Ref. 14.

Figure 3

Schematic of the incorporation velocity $\alpha (p,{\mu }^{ls}-\xi )$ and the crystallization velocity $\omega \left({\mu }^{ls}\right)$ as a function of the droplet supersaturation nanowire ${\mu }^{ls}$; $\alpha (p,{\mu }^{ls}-\xi )$ is given for two different radii, $r_0$ (gray curve) and $r$.

Figure 4

Data set 2 of Fig. 2c after Weyher (Ref. 9) and calculated fit using Eq. (13) and the following parameters: $C^s=30\mathrm{J}{\mathrm{mol}}^{-1}\mu \mathrm{m}$, $r_0=3\mu \mathrm{m}$, $v_0=615\mu \mathrm{m}{\mathrm{h}}^{-1}$, ${\omega }_1=33.6\mu \mathrm{m}{\mathrm{h}}^{-1}\mathrm{mol}{\mathrm{J}}^{-1}$, ${\alpha }_2=-0.88\mu \mathrm{m}{\mathrm{h}}^{-1}{\mathrm{mol}}^2{\mathrm{J}}^{-2}$.