Kinetic narrowing of size distribution

We present a model that reveals an interesting possibility for narrowing the size distribution of nanostructures when the deterministic growth rate changes its sign from positive to negative at a certain stationary size. Such a behavior occurs in self-catalyzed one-dimensional III-V nanowires and more generally whenever a negative “adsorption-desorption” term in the growth rate is compensated by a positive “diffusion flux.” By asymptotically solving the Fokker-Planck equation, we derive an explicit representation for the size distribution that describes either Poissonian broadening or self-regulated narrowing depending on the parameters. We show how the fluctuation-induced spreading of the size distribution can be completely suppressed in systems with size self-stabilization. These results can be used for obtaining size-uniform ensembles of different nanostructures.

Figure 1

Chart schematizing possible growth scenarios with the stationary size (attractive point $r_s=50$), regular growth, and the critical size (repulsive point $r_c=50$).

Figure 2

Binary III-V nanowires growing from a group III rich alloy droplet (e.g., GaAs nanowire growing from Ga droplet with a low concentration of As) with the vapor fluxes $v_3, v_5$ and the effective diffusion length of the group III adatoms ${\lambda }_3$; $l$ and $r$ are the nanowire length and top radius in the units of lattice spacing. The elongation rate $v_5$ is determined by the vapor flux on a nondiffusive group V element and equals the group III sink from the droplet. At a low group III influx $v_3$, the diffusion flux of the group III adatoms from the nanowire sidewalls compensates a negative III/V influx imbalance and leads to self-stabilized radial growth to the stationary radius of the nanowire top $r_s$, regardless of the initial droplet size $r_0$, as in (a). Increasing $v_3$ yields the transition to regular radial growth where the nanowire top radius increases all the way, as in (b).

Figure 3

The most representative radius in the SDs vs the mean length in the regimes of infinite growth ($a=0.01, b=5$) and radius self-stabilization to $r_s=125$ ($a=-0.2, b=25$) for two different initial conditions: $r_0=60$ in both regimes (solid lines), $r_0=190$ in self-stabilized growth, and $r_0=150$ in infinite growth (dashed lines).

Figure 4

Broadening of the SDs in the case of infinite growth for $r_0=60, {\psi }_0=2000$ (solid lines) and $r_0=150, {\psi }_0=9100$ (dashed-dotted lines); other parameters are the same as in Fig. 3.

Figure 5

Narrowing of the SDs in the case of self-stabilized growth for $r_0=60, {\psi }_0=1500$ (solid lines) and $r_0=190, {\psi }_0=13000$ (dashed-dotted lines); other parameters are the same as in Fig. 3.