ARTICLE pubs.acs.org/JPCC
Orbital-Free DFT Simulations of Elastic Response and Tensile Yielding of Ultrathin [111] Al Nanowires Linda Hung† and Emily A. Carter*,†,‡ †
Program in Applied and Computational Mathematics, and ‡Department of Mechanical and Aerospace Engineering and the Andlinger Center for Energy and the Environment, Princeton University, Princeton, New Jersey 08544, United States ABSTRACT: Elastic properties and mechanisms for tensile yielding of fcc Al nanowires are explored using orbital-free density functional theory. We quasistatically load ultrathin nanowires that have circular cross sections, diameters 1-8 nm, lengths up to ∼20 nm, and axes along the [111] direction. We find that Young’s modulus is roughly consistent for the nanowires and bulk Al but that the equilibrium interlayer spacing, elastic limit, and yield strength are diameter dependent. Plasticity is nucleated by 3-fold symmetric axial displacements of surface atoms for all nanowires examined. However, the nanowire with 4 nm diameter yields via partial slip before achieving the theoretical strength of bulk Al, while the 1 and 2 nm nanowires yield via amorphous mechanisms, above or near the theoretical strength. These results give new insight into the plastic yielding mechanisms of ultrathin fcc nanowires with edge-free cross sections and [111] orientation.
’ INTRODUCTION Nanowires behave very differently from bulk materials due to their one-dimensional nature. Novel behavior is especially pronounced in the ultrathin regime (diameters less than 10 nm). The extreme case of a linear chain of metal atoms has been studied experimentally and computationally for quantum conductance1-3 and spontaneous magnetization,4-6 and such properties may persist throughout the ultrathin nanowire regime. With unique applications like nanoelectronics7 and nanoelectromechanical systems,8,9 the structural and mechanical properties of ultrathin metal nanowires are an active area of research.10 The mechanical properties of nanowires can be examined through compression, tensile, and bending tests but ultrathin metal nanowires have rarely been tested experimentally. This is due to difficulties in synthesizing, handling, and stabilizing the nanowire, as well as the need to advance experimental techniques at the nanoscale. Tensile tests have been performed only quite recently on true ultrathin nanowires.11,12 Before that, experiments typically probed either nanojunctions up to only a few atoms in diameter1 or, at the other extreme, nanowires/nanopillars (sometimes polycrystalline) tens or hundreds of nanometers in diameter.13-28 Nevertheless, the consensus from experiments has been that the strength of nanowires is much greater than bulk materials and tends to be size dependent. While ultrathin nanowires are difficult to study in experiments, simulations are ideal for modeling their failure mechanisms down to atomic resolution. Motivated by recent experimental tensile tests with ultrathin Au nanowires (around 6-7 nm in diameter),12 here we examine the yielding mechanism of Al nanowires of similar diameter under tensile loads using orbital-free density functional theory (OFDFT), a first-principles quantum mechanical theory29,30 that can be made to scale as O(N log(N))—for some measure of system size N—even for metals.31-33 Given that Au and Al share a r 2011 American Chemical Society
face-centered-cubic (fcc) crystal structure and have sp valence bands, we anticipate that their behavior at the nanoscale could be similar, and since OFDFT has been thoroughly vetted for mechanical properties of bulk Al,34-37 we have chosen to examine Al nanowires here. We recognize that other than under ultrahigh vacuum conditions, Al nanowires in ambient conditions will develop an oxide scale that undoubtedly would alter the mechanical properties predicted here. We therefore expect the behavior predicted here to be relevant either for fcc noble metals such as Au or Pt that do not oxidize or especially for Al in the absence of oxygen. Lu et al.’s experiments12 have demonstrated two unique forms of rupture for fcc nanowires grown in the [111] crystallographic direction: dislocation-mediated necking resulting in ductile failure and twinning that produces brittle failure. Since the preference for one type of failure over the other is not understood, we model cylindrical ultrathin Al nanowires with quasistatic loading, attempting to obtain some general insights into the yielding mechanisms of fcc nanowires with [111] orientation. Al nanowires with diameters between 1 and 8 nm are examined, since ultrathin nanowires in that size regime are likely to still have an fcc structure and not be so thin that they take on helical or other unconventional structures.38-41 Although the solution-phase synthesis of ultrathin [111] nanowires has only been developed recently,42 a variety of computational studies have already been conducted to study [111] ultrathin nanowires using classical embedded-atom method (EAM) interatomic potentials. Using the modified EAM, Young’s modulus was found to barely decrease with increasing diameter for [111]-oriented Au nanowires, contrasting with a sharp drop for [110] Au nanowires.43 Au nanowires with nearly square cross sections of 2 Received: December 22, 2010 Revised: February 22, 2011 Published: March 16, 2011 6269
dx.doi.org/10.1021/jp112196t | J. Phys. Chem. C 2011, 115, 6269–6276
The Journal of Physical Chemistry C 2 or 4 4 nm2 were also modeled with the EAM, undergoing tensile loading until initial dislocation nucleation occurred at nanowire edges.44,45 While both partial and full dislocations were found in [110]-oriented nanowires, [111]-oriented nanowires only underwent partial slip while yielding. Cu and Ni [111] and [110] nanowires of similar size have also been studied with the EAM. [111] Ni was predicted to deform with disordered defects nucleating at the nanowire surface, and dislocation-mediated plasticity occurred with the [110] orientations of both materials and the [111] Cu nanowires.46 Au nanowires with hexagonal cross sections of approximate diameter 2.3 nm similarly were found to initially yield through partial slip, while later stages of yielding occurred through atomic rearrangements at the necks.47 Full yielding until fracture was also modeled with a Au nanocontact (33 layers long) with [111], [100], and [110] orientations using an EAM-like potential.48 The [111] nanocontact, whose thinnest point had a circular cross section 1.86 nm in diameter, exhibited a variety of failure mechanisms including three-plane slip, a combination of two atomic layers to form three layers of smaller diameter, and disordered yielding (some of which became reordered later in the yielding process). These previous simulations predicted some behavior for [111] nanowires distinct from expected bulk behavior, but at the same time, an interatomic description (classical interaction potential or quantum mechanics) that is not fitted to empirical bulk properties may be needed to accurately model materials with such a large surface-to-volume ratio. To this end, Kohn-Sham density functional theory (KSDFT) has also been used to model the tensile loading of ultrathin nanowires. Simulations showed that a dimer contact formed before Al nanowire fracture49 but that single-atom chains formed in the case of Au nanowires.50 Au nanowires were studied further using KSDFT by examining a variety of (111)stacked nanowires undergoing tensile stress both along the wire axis and off at an angle; linear atomic chains that broke at fairly consistent forces were found to occur in all cases.51 Unfortunately, KSDFT simulations are very expensive. The studies cited above modeled only up to ∼100 atoms, corresponding to extremely thin and short nanowires (less than 10 layers of (111) planes with no more than 10 atoms per layer). It is also unclear whether the initial geometries studied using KSDFT correspond to actual nanowire geometries, either for very thin nanowires or for atomic structures of thicker wires immediately before fracture. A method intermediate in accuracy compared to KSDFT and the EAM, the semiempirical tight-binding (TB) method, also has been used to model nanostructures under tensile loading. In a simulation of 1800 Au atoms, a multicolumnar structure appeared during yielding, and atoms initially at the nanowire surface formed a linear atomic chain before failure. However, the equilibrium configuration of the nanostructure being studied was wider than it was long, rendering results unrepresentative of a nanowire’s mechanical response and highlighting the importance of realistic initial geometries.52 A second, much smaller TB study modeled Au and Cu nanowires constructed from 80 atoms with 6-7 atoms per (111) plane. Before breaking, linear atomic chains in this simulation were nucleated by the movement of atoms that were originally in the interior of the nanowire.53 The computational advantage of TB over KSDFT is attractive, especially since TB retains some quantummechanical character. However, similar to the EAM, a good parametrization remains essential. The TB parameters in the two studies were based on bulk properties and may not provide an accurate representation of the nanowire regime. In this work, we attempt to bridge the gap between previous studies by modeling 1-8 nm diameter Al nanowires with OFDFT,
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starting from initially unstrained equilibrium structures and continuing to load them until yielding. With OFDFT, we have a first-principles quantum mechanics method that is quasilinear scaling, allowing the study of many thousands of atoms or more.31 OFDFT is more transferable than the EAM, is faster than KSDFT, and requires much less parametrization than TB. A past OFDFT study modeled [100]and [110]-oriented nanowires from 0.3 to 6 nm diameter undergoing tension and compression using the minimum periodicity in the axial direction,41 but here we perform significantly larger-scale OFDFT simulations by modeling nanowires ∼20 nm long in the periodically replicated axial direction. This length is large enough to allow plastic behavior like dislocation necking and twinning to take place without undue influence from the periodic boundary conditions. We begin by detailing the simulation setup and computational methods used to quasistatically load the [111] nanowire. A short (periodically constrained) nanowire is used when physically relevant, and a more expensive 20 nm long nanowire is used otherwise. Next, we describe nanowire behavior in the elastic regime by delimiting the range of the regime, determining the size-dependent response of nanowires, and comparing the properties of these nanowires relative to bulk Al or to nanowires with different crosssectional shapes from past work. We also discuss plastic and yielding behavior, which is also found to be qualitatively size dependent. Finally, in the conclusion, we summarize our results and discuss what these simulations for a [111] Al nanowire indicate about experimental studies of yielding in ultrathin fcc nanowires.
’ METHODS OFDFT is a quantum mechanical theory that does not use wave functions in any form but rather solves directly for the electron density distribution. One consequence of this is that density functionals must be used to calculate the kinetic energy, unlike KSDFT where a system of one-electron wave functions is constructed for this purpose. Another difference from KSDFT is that pseudopotentials (which represent the external potential arising from “ions” comprised of the core electrons and nucleus) must be spherically symmetric.29,30,32 These differences mean that OFDFT is more limited in the materials that it can accurately model compared to KSDFT. However, OFDFT’s computational efficiency makes it preferable to KSDFT for materials where it is sufficiently accurate. Indeed, because all terms can be made quasilinear scaling with a small prefactor and a parallelized code, 3-5 orders of magnitude speedup is achieved, as manifested in a recent simulation of more than a million atoms with OFDFT on a modest number of processors (
