Significance of lognormal nanocrystal size distributions

Metallic or semiconductor nanocrystals produced by very different techniques often display size distributions whose limiting shape (e.g., after long annealing times) is self-preserving and close to lognormal. We briefly survey the diverse microscopic mechanisms leading to this behavior, and present an experimental study of its inception in the case of semiconducting nanocrystals synthesized by ion implantation in silica. This example shows how the ultimate lognormal distribution is related to the system’s memory loss of initial nucleation and growth processes.

Figure 1

Illustration of the effect of implant concentration on growth characteristics. Two $\mathrm{Si}{\mathrm{O}}_2$ wafers were sequentially implanted with Pb and S at the same energies, but to different fluences (see text). $R_p$ is the projected range of Pb and S, according to the SRIM code. The $1.24\mathrm{at.}\%$ sample was annealed $1\mathrm{h}$ at $890°\mathrm{C}$ and the $0.24\mathrm{at.}\%$ sample was annealed $1\mathrm{h}$ at $900°\mathrm{C}$.

Figure 2

Experimental radius distribution functions (a) and repartition functions (b) for two samples from PbS1 (Table ) annealed respectively $1\mathrm{h}$ (inverted triangles) or $8\mathrm{h}$ (circles) at $900°\mathrm{C}$. Full (dashed) lines are lognormal (Gaussian) fits to the data.

Figure 3

Influence of annealing temperature on radius repartition function of nanocrystals from PbS1 sample grown at two different temperatures (Table ). Full lines are lognormal fits to the data.

Figure 4

Radius repartition function (dots) of different samples: PbS1 $(\mu =5.7\mathrm{nm})$ annealed $8\mathrm{h}$ at $800°\mathrm{C}$ or $(\mu =8.7\mathrm{nm})$ annealed $8\mathrm{h}$ at $900°\mathrm{C}$; PbS2 $(\mu =3.2\mathrm{nm})$; PbS3 $(\mu =4.7\mathrm{nm})$ annealed $1\mathrm{h}$ at $900°\mathrm{C}$; CdSe $(\mu =5.0\mathrm{nm})$ annealed $1\mathrm{h}$ at $900°\mathrm{C}$. Radii were normalized by the geometrical average radius. Full line, lognormal (geometrical standard deviation 1.5). Dashed lines show the 95% confidence level for the lognormal fit (population 300 precipitates).

Figure 5

Comparison (from TEM identification of nanocrystals) of PbS concentration depth dependence for samples from PbS1 after annealing at $900°\mathrm{C}$ for 1 or $8\mathrm{h}$.

Figure 6

(a) Comparison of initial concentration profiles of Pb and Te as determined by SRIM (thin lines, full and dashed) with precipitated element concentration as measured by TEM, assuming that all the precipitates are either PbTe (full dots) or Te (circles). Sample is $\mathrm{Te}+\mathrm{Pb}$ sequentially implanted silica annealed $1\mathrm{h}$ at $890°\mathrm{C}$. (b) Size repartition function in Gaussian coordinates as a function of depth in the $\mathrm{Te}+\mathrm{Pb}$ sequentially implanted silica sample, after annealing $1\mathrm{h}$ at $890°\mathrm{C}$. Two depth ranges were analyzed separately: $0–220\mathrm{nm}$ corresponds to the implanted profile (dotted line), and the other to greater depths (full line). Thin straight line, lognormal fit to nanocrystal radius repartition function for depths greater than $220\mathrm{nm}$ $(\sigma =1.4)$.

Figure 7

Reduced volume $\left(u\right)$ probability density plot. Crosses are experimental data for various semiconductor (PbS, PbSe, CdSe, PbTe) nanocrystals grown after sequential implantation of the components into pure silica (fluences ca. ${10}^{15}\text{atoms}{\mathrm{cm}}^{-2}$) at energies in the $100–300\mathrm{keV}$ range, and annealing in the range $800–900°\mathrm{C}$ for several hours. The full line is the maximum entropy distribution $e^{-u}$, determined by the sole constraint of volume conservation; the dashed line is the best fit of experimental data $(u>1)$ to the reduced lognormal distribution $(\mu =1)$.