Pattern formation in nanowires

We investigate the stability and dynamics of pine tree nanowires, i.e. nanowires which contain a screw dislocation. It induces a so called Eshelby twist of the entire wire. The dislocation is metastable in the center of the wire, and it influences the Rayleigh-Plateau instability, which favors decomposition of a long wire into droplets.

Introduction

*Fig. 1: Schematic decomposition of a nanowire into droplets with a preferred wavelength.*

© Rainer Schulz, Max-Planck-Institut für Eisenforschung GmbH

*Fig. 1: Schematic decomposition of a nanowire into droplets with a preferred wavelength.*

© Rainer Schulz, Max-Planck-Institut für Eisenforschung GmbH

Nanowires have fascinating electronic and mechanical properties due to their two-dimensional character with huge industrial applicability potential. Their diameter is typically below 100 nm, and they posses a large aspect ratio between length and radius of more than 100, exhibiting one dimensional properties e.g. in quantization of the electrocis states. Nanowires belong to the best controllable structures on the nanoscale. Potential applications are field effect transistors, p-n diods, LEDs, nanolaser, complex logical gates, gas sensors, nanoresonators, etc.. Since their properties can be well controlled during fabrication, they can be designed with specific electronic properties.

From a technological point of view, he mechanical stability of the wires on long timescales is essential. As a general fact, cylindrical objects suffer from an instability towards a decomposition into droplets, driven by isotropic surface energy. This instability has been predicted by Rayleigh and Plateau, and has been investigated by Nichols and Mullins. It can play a substantial effect also in nanowires. This predicts that sufficiently long wires will always decompose into droplets. The growth rate for such an instability reaches a maximum for a specific wavelength of the perturbation, which depends on the transport mechanism, like surface diffusion. There are experimental observations for the decomposition of gold nanowires at temperatues below 500 C. Due to the low temperatures where this degradation process can occur, it has to considered for the production and use of nanomaterials. On the other hand, such natural instabilities may be used on purpose for applications in optical conductors.

Pine tree nanowires

Fig. 2: The screw dislocation is only metastable in a nanowire with the Eshelby twist and can therefore move out if it is located in the outer region of the wire. Simulation snapshot on the top view of a wire, using amplitude equations.

© Robert Spatschek, Max-Planck-Institut für Eisenforschung GmbH

Fig. 2: The screw dislocation is only metastable in a nanowire with the Eshelby twist and can therefore move out if it is located in the outer region of the wire. Simulation snapshot on the top view of a wire, using amplitude equations.

© Robert Spatschek, Max-Planck-Institut für Eisenforschung GmbH

Pine tree nanowires have a tree-like branched structure. The wire has in its trunk an axial screw dislocation, whereas the side branches are typically free of defects. The screq dislocation induces a torsion of the wire due to a torque - know as Eshelby twist. Eshelby has calculated the mechanical energy of the wire as function of the position of the dislocation line in an infinitely ling wire and found that the position in the center is only metastable. The Eshelby twist has been observed in pine tree nanowires, and the torsion angle agrees with the theoretical predictions.

*Fig. 3: Energy of the twisted wire as function of the position of the screw dislocation. The center position is only metastable.*

© Rainer Schulz, Max-Planck-Institut für Eisenforschung GmbH

*Fig. 3: Energy of the twisted wire as function of the position of the screw dislocation. The center position is only metastable.*

© Rainer Schulz, Max-Planck-Institut für Eisenforschung GmbH

We have used analytical calculations and finite element computations to investigate the energetics of pine tree nanowires, as well as their long term decomposition based on the Rayleigh-Plateau instability. This leads to a combination of surface energy induced morphological changes together with mechanical deformations due to the twist. The tangential stresses at the wire surfaces can therefore accelerate the decomposition in the sense of a Grinfeld instability.

Coherent nanowires

The physics of nanocrystals deposited at the surface of a substrate are of wide interest, owing to the fundamental physical questions posed and their large spectrum of industrial applications. Although the morphology of nanocrystals can depend on the history of the production process, for example for a catalyst-assisted growth, it may be estimated by an energy minimizing procedure. Epitaxial crystalline nanostructures elongated perpendicular to their supporting substrate such as free-standing nanowires or nanoneedles, have attracted a great attention in the last years owing to their ability to sustain a coherent interface with the substrate even for large misfits. The equilibrium shape of such crystals depends crucially on whether they are coherently attached to the substrate or not, owing to the very different corresponding elastic interactions.

For elongated structures perpendicular to the substrate, the coherency at the crystal/substrate interface is sustained due to its finite area. For an infinite contact area with the substrate, the critical height (perpendicular to the substrate) of the crystal above which dislocations are energetically favored is about 5–10 nm for misfits of the order of 1%. For a finite linear dimension of the coherent interface, the critical height depends on this value.

In this work an energy minimizing procedure is performed for a crystal of a certain volume having a cylindrical geometry elongated perpendicular to the misfitting substrate, i.e. a nanowire. Qualitative scaling laws are derived for the dependence on the volume of the linear dimension of its interface with the substrate and of its height.