Journal Club Theme of November 2014: Catalyzed Growth of Semiconductor Nanowires - Multiscale Models

Introduction

Happy Halloween!

In this issue of the Journal Club, I would like to introduce the problem of catalyzed growth of semiconductor nanowires and ways to model it.  I would provide web links to only a few articles, hoping that you would download and actually read them so that we can have a more in-depth discussion. 

Semiconductor nanowires are semiconductor materials with dimension less than 100 nm in two dimensions and their length can extend to multiple microns in the third dimension.  Due to their geometry, they have many unique electrical, optical and catalytic properties and have many promising applications, e.g. in integrated circuits, solar cells, batteries, biosensors.  For example, semiconductor nanowires can be used to make field-effect transistors (FETs) in which the gate can wrap around the entire nanowire, offering better performance than planar FET structures.  Surround-gate nanowire FETs are recognized in the International Technology Roadmap for Semiconductors as a critical enabler for future dimensional scaling in electronics.

To enable nanowires in devices, we need to be able to synthesize them in a controlled way.  In general, there are two ways to synthesize semiconductor nanowires that are compatible with the existing semiconductor industry: the top-down approach and the bottom-up approach.  The top-down approach means etching from a patterned mask.  There are limitations on how small the nanowire features can be obtained through etching.  The bottom-up approach means growth from a catalyzed nano-particle, and is the focus of this discussion. 

Suggested reading: Semiconductor nanowire fabrication by bottom-up and top-down paradigms, R. G. Hobbs, N. Petkov, J. D. Holmes, Chemistry of Materials, 24, 1975-1991, 2012, dx.doi.org/10.1021/cm300570n

The vapor-liquid-solid (VLS) process is the most widely used method for controlled growth of semiconductor nanowires.  Since it is the focus of this discussion, here is a brief introduction of the VLS process.

As an example, consider a gold nano-particle deposited on a silicon substrate.  By heating the substrate, the nano-particle melts and form a gold-silicon (eutectic) liquid droplet.  At the same time, we introduce a Si containing gas, such as silane (SiH4).  The silane molecule decomposes at the surface of the liquid droplet.  Hydrogen escapes as a gas and silicon enters the liquid droplet, leading to a supersaturation of silicon in the liquid.  By increasing the gas pressure, it is possible to have sufficient supersaturation that pushes silicon to condensate at the bottom of the droplet.  The solid phase grows up in the form of a nanowire with the catalyst droplet remaining on top.  The process is called VLS because all three phases (vapor, liquid, solid) are involved.  Silicon atoms travels from the vapor phase, through the liquid phase, and finally enters the solid phase (the nanowire).

An advantage of the VLS growth is that the diameter of the nanowires can be conveniently controlled by the size of the catalyst particle (which can be very well controlled by chemical means).  However, there are also growth anomalies, such as kinking.

(Image courtesy of Prof. Paul McIntyre, Stanford University)

Obviously, growth anomalies need to be avoided if we want to make nanowire devices reliably.  But the origin of such growth anomalies is not well understood.  Hence we hope modeling can be of help.

Atomistic model

A full atomistic model is probably a natural place to think about this problem.  After all, the dimension of the nanowire is, of course, at the nanoscale.  It is possible to model all the atoms involved in the VLS growth process.  A few years ago, we started to develop an atomistic (e.g. molecular dynamics) model of the VLS growth process.  We chose the Si-Au as our model system, and realize that we need a reasonable interatomic potential for Si-Au to get started.  We ended up developing a Si-Au potential ourselves (despite the advice I received when I was a graduate student from my former advisor to stay away from the potential developing business).  What we learned is that, indeed, developing a interatomic potential is a very tricky business.  Nonetheless, my former student, Seunghwa Ryu (now an assisitant professor at KAIST) persisted and succeeded in developing a Si-Au potential that is reasonably well fitted to the Si-Au binary phase diagram.  Well, it turns out to be extremely difficult to fit the eutectic composition (19%) of the phase diagram.  We did the best we could.  Anyway, I am attaching the phase diagram below.  You can judge for yourself how good is our fit.

For more details see: A gold–silicon potential fitted to the binary phase diagram, Seunghwa Ryu and Wei Cai, J. Phys.: Condens. Matter 22, 055401 (2010) doi:10.1088/0953-8984/22/5/055401

Using this potential model, we performed a set of molecular dynamics simulations of gold catalyzed growth of silicon films and nanowires.  For example, we found indication that growth of (111) surface proceeds in a layer-by-layer fashion while growth of (110) surface does not.  This is consistent with the experimental observation that <111> oriented nanowires tend to grow more slowly than <110> oriented nanowires. 

For more details see: Molecular Dynamics Simulations of Gold-Catalyzed Growth of Silicon Bulk Crystals and Nanowires, Journal of Materials Research, 26, 2199 (2011).  doi:10.1557/jmr.2011.155

Unfortunately, we weren’t able to get much mileage out of the atomistic model.  This is mostly due to the extreme limit of the time scale of molecular dynamics (MD) simulations.  Because 1ns is already pretty long for MD simulations, the nanowire growth speed in our MD simulations is of the order of 1 nm/ns.  On the other hand, the experimental growth speed is on the order of 1 nm/s.  That’s a gap of nine orders of magnitude, which is hard to cross.

Continuum model

To make progress, we decided to construct a continuum model for VLS growth, which (I hope) is the focus of this issue of Journal Club discussion.  (Sorry it took so long to get to this point.)  By using a more coarse-grained model, not only we are able to model systems that are larger in size, but also (more importantly) the model tends to have much longer time scale as well.

We adopted the so-called “multi-phase field” formulation.  The system is described by three phase fields: phi_V, phi_L, phi_S, corresponding to the vapor, liquid, and solid phases, respectively.  For each i = V, L, S, phi_i (x) = 1 means that point x is occupied by phase i, while phi_i (x) = 0 means that point x is not occupied by phase i (see figure below).

The phase fields evolve in a direction that reduces the total free energy functional.  We performed some benchmark tests, e.g. to reproduce the Young’s angle at triple phase junctions when simulating an equilibrium condition (see below). 

This turns out to be an important self-consistency check for the phase field formulation.  Actually we had difficulty reproducing the Young’s angle in an earlier formulation of the phase field model, and we eventually learned that it was because the formulation was not thermodynamically self-consistent.  As a result, we had to abandon that formulation (even thought it produced nice movies of nanowire growth) and eventually settle down with the formulation that we are using now. 

When the chemical potential of the vapor phase is set to be higher than that in the solid phase, the phase field model generates nanowire growth.  We believe this is the first 3D phase field for VLS growth, and I hope I have provided enough motivation for you to read the paper.

Suggested reading: A three-dimensional phase field model for nanowire growth by the vapor-liquid-solid mechanism, Model. Simul. Mater. Sci. Eng. 22, 055005 (2014), doi:10.1088/0965-0393/22/5/055005  (also check out supplementary information for more goodies)

Here are some more reasons if you are still debating whether or not to download and take a look at the paper.  This phase field model, despite its simplicity, actually captures a lot of the interesting (geometric) features of VLS nanowire growth.

From the 3D snapshot above (nanowire in a box), we can see that the cross section of the nanowire has a hexagonal shape.  However, the cross section is not a regular hexagon.  The hexagon consists of three long edges and three short edges.  The side surfaces of the short edges are actually zig-zagged.

The irregular hexagonal shape of the nanowire cross section agrees very well with experimental observation (see below).

For the same reason, if we take the side view, the left side and right side of the nanowire are not symmetric, as shown below.

OK, I hope I have piqued your interests sufficiently that you will be motivated to read our paper.  I look forward to our discussions.