Structures of Medium-Sized Silicon Clusters

4/9/1998

#silicon clusters #computational physics #genetic algorithm #ion mobility #materials science

612 citations (CrossRef). A Nature paper and, by citation count, one of the most impactful papers I have been involved with — predating my particle physics career.

The Problem

Silicon clusters (Si_n) are the building blocks of semiconductor nanotechnology, but their structures were unknown for sizes beyond a handful of atoms. Unlike carbon clusters, which form well-known structures like fullerenes and nanotubes, silicon clusters do not simply adopt fragments of the bulk diamond cubic lattice. Theoretical predictions of cluster geometries proliferated, but without experimental validation there was no way to determine which structures nature actually preferred. The challenge was twofold: computing the vast landscape of possible geometries for a given cluster size, and matching the predictions against experimental measurements sensitive to shape.

The Approach

The paper combined a genetic algorithm for global geometry optimization with ion mobility spectrometry for experimental validation. The genetic algorithm searches the potential energy surface by evolving a population of candidate structures, selecting for low energy and good fitness, avoiding the trap of local minima that plague conventional optimization. Ion mobility measurements, performed by Martin Jarrold’s group, measure how quickly a charged cluster drifts through a buffer gas — a quantity directly related to the cluster’s cross-sectional shape. Kai-Ming Ho’s group computed the theoretical ion mobilities for each predicted geometry using trajectory calculations, and the comparison between theory and experiment was definitive.

The Key Results

For Si₁₂ through Si₁₈, the structures are built from stacked Si₉ tricapped trigonal prisms — a motif that has no analog in bulk silicon. For Si₁₉ and larger, near-spherical cage structures emerge as the most stable forms. The determined geometries do not correspond to bulk fragments at any size studied, a significant departure from the expectation that clusters would simply be small pieces of the diamond lattice. The agreement between computed and measured ion mobilities was excellent, providing the first reliable structural determination for silicon clusters in this size range.

Recollections

I was an undergraduate at Iowa State, and I had taken Kai-Ming Ho’s thermal physics course taught out of Kittel and Kroemer. I hadn’t taken any quantum mechanics, which wasn’t a formal requirement, but to say I was out of my depth would be an understatement. I worked incredibly hard on that course, struggling with Fermi-Dirac and Bose-Einstein statistics that I didn’t understand going in. After the semester was over and I’d done reasonably well, Ho saw me in the hallway and offered me the chance to work with his group.

Ho had pioneered using genetic algorithms in conjunction with simulated annealing to find the ground states of N-body carbon molecules, showing that many carbon clusters produced gorgeous shell structures. He was now extending the technique to silicon. The practical application was direct: if you could identify especially stable silicon structures, those might be what would break off during silicon wafer manufacturing. Understanding the clusters had real industrial relevance.

When I sat down to look at the outputs of their computational runs, they had these blobs of 14, 15, 16 atom silicon clusters, over a thousand of them from each run. They all looked similar but were subtly different. The group was spinning them around on their desktops trying to classify them visually. You couldn’t easily use average binding energy because a small dislocation of a single atom would spike the energy of an otherwise nearly identical structure. I was pretty lost looking at these things.

I figured there had to be a way of classifying the structures without manual inspection. I realized that if you computed the interatomic distances of all pairs of atoms, that would produce a spectrum that was rotationally and translationally invariant: rotate or translate the molecule and the spectrum stays the same. And if a single atom was mildly displaced, the spectrum would change only slightly rather than catastrophically. I defined a function f(d) as the number of interatomic distances smaller than d for each molecule, then computed the integral of |f_i(x) - f_j(x)| dx to get a distance metric between any two structures. This let us take the thousand-plus molecules from each run and quickly identify the distinct ground states. In those days there was no Python, and few of the standard C++ libraries that would make this trivial today. It ended up being a few thousand lines of code, a substantial programming problem for an undergraduate. But it worked, and it let the group extend the analysis out to clusters of a few dozen atoms. My contribution wasn’t the most important element of the science in the paper, but it was useful, and I was happy to find out later that I had been included as an author on a Nature paper.