Single crystalline ultrathin gold nanowires: Promising nanoscale interconnects

Using first principles based density functional calculation we study the mechanical, electronic and transport properties of single crystalline gold nanowires. While nanowires with the diameter less than 2 nm retain hexagonal cross-section, the larger diameter wires show a structural smoothening leading to circular cross-section. These structural changes significantly affect the mechanical properties of the wires, however, strength remains comparable to the bulk. The transport calculations reveal that the conductivity of these wires are in good agreement with experiments. The combination of good mechanical, electronic and transport properties make these wires promising as interconnects for nano devices.

The trend towards reducing the size of electronic devices has increased the demand for developing suitable interconnects with metallic conductivity at the nanoscale. Resistance to oxidation and excellent electrical conductivity are two critical attributes that makes gold an ideal interconnect material.¹ Among the various nanostructures of gold, nanowires are best suited as interconnects owing to their 1D geometry and superior conducting^2–4 properties. Furthermore, gold nanowires (AuNWs) have also been shown to be promising for photonic,⁵ plasmonic⁶ and sensing^7–11 applications.

Experimentally synthesized gold nanowires (AuNWs) are in general polycrystalline¹² and typically are of larger diameters. These polycrystalline AuNWs are synthesized by assembly of stable atomic clusters.^13,14 On the other hand, the thinnest possible nanowire of Au is a single atomic chain (SAC), fabricated^15,16 under STM tip at 4.2 K. While, the polycrystalline large diameter AuNWs (∼50 nm) exhibit excellent mechanical strengt,¹⁷ grain boundaries in such nanowires act as scattering centres, thereby adversely affecting its transport properties and electromigration further limits its conductivity.^18,19 On the other hand, the SAC have better transport properties;¹ however, it suffers from poor mechanical strength.

Until recently, AuNWs in the intermediate diameter range were elusive to experimentalist due to several synthetic challenges like template removal and less control over growth due to lack of proper capping agents. Single crystalline [111] direction oriented, AuNWs of ∼2 nm diameter²⁰ have been recently synthesized chemically via oriented attachment of crystalline Au-nanoparticles. The diameter of these nanowires depends upon the size of the nanoparticles. These single crystalline nanowires have very low concentration of extended defects compared with the polycrystalline NWs. Furthermore, as the quantum confinement effects start to play role at the nanoscale, electronic properties of these nanowires can be radically different from their bulk counterparts.²¹ The diameter range of these AuNWs is particularly exciting for interconnects in nanodevices. This demands a thorough study of mechanical and transport properties of such single crystalline AuNWs. Although molecular dynamics(MD) simulation has been carried out^22–25 to study the mechanical properties of larger diameter AuNWs, the results of these calculations depend on the interaction potential used. Furthermore, to the best of our knowledge, there is no systematic ab initio study on mechanical and electronic properties of [111] oriented bulk AuNWs. Here we show using first principles calculations that the single crystalline AuNWs are metallic and mechanically stable at nanoscale. These AuNWs exhibit diameter dependent shape transformation, which influences the mechanical properties significantly. The wires exhibit comparable mechanical strength and lower DC conductivities compared to bulk Au. However, the conductivity values are still high enough for them to serve as potential interconnects.

The calculations were performed using first principles density functional theory. The ionic cores are described by all-electron projector augmented wave potentials^26,27 and the Perdew-Burke-Ernzerhof²⁸ generalized gradient approximation to the electronic exchange and correlation as implemented in the Vienna Ab Initio Simulation Package (VASP) code.^26,27,29 Au is a heavy element and therefore, the scalar relativistic effects are incorporated in the PAW pseudopotential. Geometries were considered converged, when the component of interatomic forces were less than 0.005 eV/ Å. The Brilluoin zone integration was performed at a Monkhorst-Pack³⁰ k-grid of 1 × 1 × 3. The cell length along the nanowire axis and the atomic positions in the unit cell have been optimized without any symmetry constraint. In the transverse directions, more than 12 Å of vacuum space was included to avoid interactions among the periodic replicas of the nanowire.

The structures of AuNWs are chosen similar to the experimentally synthesized AuNWs.²⁰ These [111] direction oriented AuNWs, were generated by cleaving bulk Au with the low index surfaces. The wires were terminated with six {110} surfaces, with hexagonal cross-sections. In order to systematically study the diameter dependent electronic and mechanical properties, five different average diameters AuNWs were considered, and denoted by NW1, NW2, NW3, NW4, and NW5, as shown in Figure 1(a). The equilibrium diameters (number of atoms in a supercell) of NW1-5 are 12.84 (61), 15.42 (91), 19.62 (169), 21.94 (217), and 23.12 (271) Å, respectively.

The equilibrium periodicity of the supercell of the AuNWs along the axial direction were optimized by varying the length along the orientation (z) of the nanowires. The energy vs. z gives a parabola, whose minimum is the equilibrium z-periodicity (z₀) of the nanowire. Figure 1(b) shows the variation of z₀ of all the nanowires as a function of diameter. The z₀ increases with increasing wire diameter. A discontinuous jump in z₀ was obtained by going from NW3 to NW4. NW4 and NW5 possess much larger equilibrium z₀ compared with the rest of the nanowires. The z₀ of NW1, NW2, NW3 are very close to the bulk value. The sudden increase in z₀ for NW4 and NW5 was caused by smoothening of the corners of hexagonal cross-section. The atoms at the corners move inwards causing a overall expansion in z₀. These geometries resemble very closely with the reported experimental crystalline nanowires,²⁰ which also have circular cross-sections.

In order to estimate the influence of the structural transformation on the mechanical properties of these AuNWs, we calculated the elastic moduli as a function of the diameter. The wires were subjected to compressive as well as tensile strain along the axial direction. The corresponding developed stress, σ_z is given by:

where, ε_z and ∂U are the strain, and change in the internal energy, respectively. V₀ is the equilibrium volume of the wire given by πr²z₀, where, r and z₀ are radius and equilibrium lattice periodicity of the wire, respectively. The radius of the wire was defined as the radius of the largest cylinder that can be inscribed in the nanowire. Within the elastic limit, the Young's modulus (γ) is given by

$(∂2U/∂εz2)$ can be obtained from a second order polynomial fit of the U vs. ε_z curve. The γ of the AuNWs as function of diameter are plotted and shown Figure 1(c). γ decreases with increasing diameter for NW1, NW2, and NW3. This trend changes drastically going from NW3 to NW4 as the γ starts to increase with the increasing diameter and appears to converge to the bulk value Figure 1(c). This sudden drop followed by an increase in γ is caused by the cross-section smoothening of the NWs, which essentially influences the mechanical strength of these larger diameter AuNWs.

The larger values of γ for the smaller diameter nanowires is not surprising. Going down the scale from bulk to nano, two factors, the volume of the wire (V₀) and the stress per unit strain (⁠

$∂2U/∂εz2$⁠) tend to modify the Young's modulus (γ) in opposite ways. The V₀ goes on decreasing, which favors the increase of the γ. Such hardening is known for nanostructures³¹ and the reason is generally attributed to the exhaustion of volume, which imparts more rigidity to the system. From NW1 to NW3 this trend is evident from Figure 1(c). For NW4 and NW5, the transformation to a circular cross-section leads to a huge surface relaxation. The resulting structure is very stable demanding very high stress for small unit strain, i.e. a high (⁠

$\partial ^{2}U/\partial \epsilon _{z}^{2}$

$∂2U/∂εz2$⁠) value. Thus, these two oppositely behaving factors makes the γ pass through a minimum. The critical point observed in our calculation incidentally coincides with structure smoothening limit. Overall the value of γ obtained here are closer (larger for the smaller diameter wires) to the bulk value, implying that the mechanical stability of these single crystalline nanowires will remain intact at nanoscale.

Furthermore, elongation or compression of a wire from its equilibrium energy configuration along the axial direction is expected to decrease or increase the cross sectional area in the lateral directions, respectively. Using change in radius as a function of the applied strain, the Poisson's ratio (ν) of the AuNWs were calculated, which is given by

where, ε_r is the strain along radial direction given as (r-r₀)/r₀, where r and r₀ are instantaneous and equilibrium radius, respectively. The ε_r vs. ε_z curves are linear, with a negative slope, which gives the Poisson's ratio.

Figure 1(c) shows the variation of the Poisson's ratio (ν) of the AuNWs as a function of diameter. Although there is no systematic trend in the behaviour, the smaller AuNWs have lower Poisson's ratio (ν), whereas it saturates to bulk value for larger diameter wires. The smoothening due to relaxation for larger diameter wires give rise to such stability. However, the smaller diameter AuNWs lack of such Poisson's ratio (ν) due to lack of high surface relaxation. Overall these AuNWs have comparable (sometimes better) mechanical properties as compared to bulk Au.

Having shown the AuNWs unusual but comparable mechanical strength to the bulk, we next study the electronic properties as in order to be a potential interconnect it must exhibit the metallic conductivity. The partial and total density of states (DOS) of AuNWs were studied as a function of diameter and compared with the bulk as shown in Figure 2. Unlike molecular DOS, there were not many sharp and discrete peaks observed in the DOS of the AuNWs. The levels overlap with each other and form a continuous electronic band, indicating that the electronic properties will be closer to the bulk Au. The shapes of the total DOS of NW1-5, are almost independent of the diameter. With the increasing diameter, the width of the bands get narrower. This is caused by the quantum confinement effect as the broadening of the bands are inversely proportional to the dimension of the 2D-well. The DOS evolves rather quickly, as from the NW3 onwards the width of the bands already becomes closer to the bulk.

To further investigate the electronic properties, atom projected electronic density of states were also studied, Figure 2. Local DOS is calculated by projecting the electronic wave function of the wire onto the spherical harmonics in the sphere around the considered atom. For every wire, one inner and one outer atom were chosen to calculate the atom projected DOS as a function of diameter of the wires. As seen from the Figure 2, the projected DOS on inner atoms expectedly resembles the projected DOS of the bulk. The states below (above) the fermi level originates from s (p and d) orbitals. The projected DOS on outer atoms are shifted downwards compared with the inner atom. With the increasing diameter these states shift upwards and resemble close to the DOS of outer atom. This gradual shift is observed for all the s, p, and d orbitals as shown in Figure 2. Therefore, the conduction of the electrons will happen mostly via inner atoms, as they contribute most to the states closer to the fermi level. Furthermore, these results also indicate that the electronic properties will not be affected much by surface relaxation or the structural smoothening as observed in the thicker wire. Overall the DOS of the AuNWs keep more or less bulk like characteristics, an indication of having not very different conductivity than bulk. Furthermore, reduction of size could also lead to magnetism, however, the spin-polarized calculations do not show any magnetism in AuNWs.

A finite DOS at fermi level, does not always translate into good conductivity. Besides metallic DOS, there are other factors, specifically scattering, which can alter the conductivity significantly. In order to investigate its effect on transport properties, the electrical conductivity of AuNWs is calculated using Boltzmann Transport theory (BTT) within constant scattering time approximation (CSTA),^32,33 as implemented in the BoltzTraP program.³⁴ In CSTA it is assumed that the scattering time τ determining the electrical conductivity does not vary strongly with temperature. In the BTT, the motion of an electron is treated semi-classically. The group velocity of an electron in a particular band can be described as

where

$ε(i,k)$ and

$\textbf {k}_\alpha$

$kα$ are i^th energy band and α component of wavevector k, respectively. From group velocity

$\nu _\alpha (i,\textbf {k})$

$να(i,k)$ the conductivity tensors can be obtained as

where

$τ(i,k)$ is the relaxation time, α and β are the cartesian indices, N is the total number of k points sampled, and e is the elementary charge. The electrical conductivity tensor is given by an equivalent equation:

where V and f_μ are the volume of the unit cell and Fermi Dirac distribution function, respectively. For calculation of transport properties first, the group velocities are obtained by Fourier interpolation^35,36 of the band energies as a function of k. Subsequently the transport properties are calculated using equation (6).

We tested the method by computing the electronic conductivity of bulk Au. Drude model relaxation time³³ is used for electrical conductivity calculation for bulk gold as well as for AuNWs. The conductivity σ of bulk Au at room temperature is found to be 1.10 × 10⁷ Ω⁻¹m⁻¹ as shown in inset of Figure 3, which agrees well with experimentally reported value³³ under CSTA. In order to estimate transport property of AuNWs, the electronic structure calculations were performed at a denser grid of 1 × 1 × 40. As expected, all the AuNWs show metallic conductivity, which decreases with increasing temperature and saturates as shown in Figure 3. The conductivity of the nanowires decrease compared with the bulk Au. This decrease is caused by the surface scattering of the carriers. The saturated value of the conductivity of the nanowires does not depend on the diameter. All the wires have almost similar conductivity 2∼4 × 10⁵ Ω⁻¹m⁻¹. This observation is in well agreement with recent resistivity measurements.^21,37 This is not surprising as the DOS of the nanowires were also very similar and almost independent of the wire diameter. Furthermore, the magnitude of the conductivity is also not very drastically different from the bulk value. Almost diameter independent conductivity imply that the single crystalline AuNWs are indeed a promising candidate for interconnects in nanoscale devices.

In summary, the mechanical and electronic properties of [111] oriented single crystalline AuNWs have been studied as a function of diameter using first principles calculations. We have shown the important role played by the structural transformation in determining the mechanical attributes of AuNWs. The AuNWs undergoes hexagonal to circular crossection transformation with the increasing diameter. This transformation affects the mechanical properties significantly. The elastic moduli as function of diameter exhibit an initial decrease followed by an increase. The values of the elastic moduli are close to the bulk Au, indicating higher stability of these single crystalline AuNWs. The AuNWs possess the finite density of states at the fermi level. The shape and characteristics of the DOS of the AuNWs closely resembles the bulk Au. Furthermore, the conductance in these wires will be through the bulk like inner atom and would rather be insensitive to the relaxation of the surfaces. All the nanowires exhibit metallic conductivity, with very little dependence on the diameter. Although due to nano structuring the magnitude of the conductivity decreases compared with the bulk, the wires remain metallic with reasonably high conductivity down to few angstrom diameter level. The bulk like mechanical stability and electronic conduction at nanoscale make these single crystalline AuNWs promising as interconnects for nanoscale devices.