Structural information of the pure Au (Au-p) and hydrogen (Au-H) or carbon contaminated (Au-C) NWs at the equilibrium and close to rupture situations. In the first row, Le is the equilibrium length of the NW, whereas L* corresponds to length just before breakage occurs. Refer to Fig. 11 for an illustration of each of the geometries analyzed. The d (1,2) bond distance and the αX bond angle of the impurity are in boldface.
“The era in which the number of transistors on a computer chip doubles at a constant rate is drawing to a close”. This is not the prophecy of an obscure mind, but is more or less the conclusion drawn by none other than the man who coined Moores´ law Times Online September 19, 2007, http://technology.timesonline.co.uk/tol/news/tech_and_web/article2489053.ece
Times Online September 19, 2007, http://technology.timesonline.co.uk/tol/news/tech_and_web/article2489053.ece
It still is far from clear which will be the technological procedure for the massive production of these molecular circuits. However, there are a number of experimental techniques for the study of their properties that are well established. These are shown schematically in Fig. 1. Fig. 1d shows a method devised to study the structure of monatomic nanowires (NWs). It has been developed by Kondo and Takayanagi (Kondo & Takayanagi, 1997) using
Another procedure that has been used to study the properties of monatomic metal contacts is the so-called
In the case of metallic
Finally, the method developed Haiss et al. moves more into the spirit of bottom up nanostructuring (Fig. 1e). In this procedure, a molecule bridges spontaneously the gap between an STM tip and a surface (Haiss et al., 2004, Haiss et al., 2006). The substrate-tip connection is verified by jumps in the tunnelling current measured.
2. Structure and stability of pure and contaminated metallic monatomic nanowires
2.1. Experimental measurements
The conductance measured through metallic monatomic nanowires is quantized in units of
Due to the inherent irreproducibility of the contacts formed, and therefore of the measured conductance curves, it is useful to construct histograms with a few hundred measured curves. A force histogram for the Au nanowire showed a narrow distribution centered at 1.5 ± 0.3 nN for the force needed to break one single bond in the chain (Rubio-Bollinger et al., 2001).
Although gold has been the most prominent metal studied so far, mechanical elongation has shown to be useful to fabricate nanowires of Ag (Rodrigues et al., 2002), Co, Pt (Rodrigues et al., 2003), Pd (Kiguchi & Murakoshi, 2006, Rodrigues et al., 2003) and Cu (Bettini et al., 2006), as well as for Au and Ag alloys of varying compositions (Sato et al., 2006b).
Two striking experimental features have drawn the attention of the researches in the last few years: first, the existence of unusually large Au – Au distances in the order of 3.6 – 4.0 Å (Kizuka, 2008, Koizumi et al., 2001, Ohnishi et al., 1998, Sato et al., 2006b), longer than the bulk distance of 2.88 Å. These experimental observations along with the simulation of HRTEM measurements suggest that the atomic-sized wires are complexed with light elements (such as H, C, S for example) (Kizuka, 2008, Koizumi et al., 2001).
Second, the long-term stability of gold nanowires at room temperature, in many cases in the order of seconds (Legoas et al., 2002, Ohnishi et al., 1998; Rodrigues & Ugarte, 2001a, Rubio-Bollinger et al., 2001,; Takai et al., 2001, Yanson et al., 1998), which is an extremely long time in relation to those characteristic of molecular motion. For instance, Fig. 2 shows the time evolution of gold chain under elongation, which is in the order of ~1 second.
The stability and breakdown of gold nanojunctions at different stretching rates has recently been measured by Huang et al. (Huang et al., 2007a). Information about the lifetime of the Au – Au bond is extracted from the length of the last plateau in the conductance traces of Au – Au point contacts. The most probable stretching distance for this last plateau (corresponding to an atom-sized contact between the electrodes as verified by its conductance at a
In Eq. 1
2.2. Computational simulations
Experimental observations such as those described above, have triggered numerous theoretical approaches to gain further insight into the structure and transmission properties of these metallic nanowires. The mechanical structure and evolution of a tip-surface contact has been modelled by means of different computational techniques. In early works,
Tight binding MD simulations using the Naval Research Laboratory potentials, along with
The main drawback of the MD simulations mentioned above is that the stretching rate used (typically in the order of 1 -2 m/s) is around 9 - 10 orders of magnitude larger than the experimental values (in the order of a tenths to a few nm/s).
The use of
As it was mentioned above, one of the possible explanations for the rather long Au – Au distances observed in a monatomic neck is the presence of light weight elements, such as C, H, O or S, intercalated between gold atoms. This possibility has only been investigated so far by the means of computational simulations, since these light weight elements would have a low contrast against the much heavier Au atoms, and thus rendering their direct visualization by today’s electronic microscopes very difficult. Carbon is a frequent contaminant in bulk gold (Legoas et al., 2002), while hydrogen (O'Hanlon, 2001) and oxygen (Bahn et al., 2002) are impurities very difficult to extract even in the best UHV conditions. Although the possibility of hydrogen acting as a contaminant has been ruled out by an AIMD study (Legoas et al., 2004), this result has been subsequently challenged (Hobi et al., 2005).
Legoas et al. (Legoas et al., 2002) modelled monatomic gold chains contaminated with carbon by means of geometry optimization at DFT-LDA level. The authors used an isolated linear chain and no tension was exerted on the system. Their results showed that long Au – Au distances of around 4 – 4.5 Å could be explained by the presence of two consecutive carbon atoms (C2) inserted into the gold chain. Whereas, another set of anomalously long bond in the order of 3 – 3.7 Å could be a consequence of a mixture of pure Au – Au bonds with contamination of such a bond by a single carbon atom.
Skorodumova and Simak (Skorodumova & Simak, 2003) showed, using DFT-GGA calculations, that the unusual structural stability of monatomic gold wires could be explained in terms of hydrogen contamination. Stretching the nanowire, the authors observed that the chain takes a linear structure with hydrogen atoms intercalated and a Au – Au distance of 8.8 Å before the nanowire breaks. The cohesive energy of the contaminated gold wire was found to be 2-fold higher than a pure chain of gold atoms. This last result was attributed to a partial charge transfer from gold to hydrogen. Subsequently, the influence of carbon was explored using the same computational methodology (Skorodumova et al., 2007, Skorodumova & Simak, 2004), also finding that carbon can enhance the stability of linear gold chains yielding large interatomic distances.
Novaes et al. (Novaes et al., 2003) studied through
The main drawback of the procedures presented so far is that the presence of the impurity is simply assumed, and no description is obtained of how, when, or with what probability it migrates to the position it was assumed to have. To overcome this limitation, an AIMD study of the formation and growth of gold chains with a variety of impurities (H, C, O, S), without any assumption of their initial positions was performed (Anglada et al., 2007). One or two impurity atoms were introduced randomly in an amorphous column of 50 – 150 gold atoms. These amorphous solid columns were stretched during 4 – 18 ns until they broke. Hydrogen was always found to evaporate before formation of the monatomic chain took place. Carbon and oxygen were found in the final chains with low probability (~ 10 %), while sulphur was found participating in it with a high probability (~ 90 %). The mean distances between gold atoms bridged by C, O and S were 3.3, 3.4 and 5.0 Å, respectively, in good agreement with experiments.
Inasmuch as this last study provides a level of accuracy and reliability superior to those mentioned before, the stretching rate is still much higher than those typically used in experiments, and the simulations last only a few nanoseconds, while experiments take place in the order of 0.001 – 1 second.
The study of the effect of impurities on the structure and stability of gold nanowires is an ongoing investigation topic. Some of the most recent theoretical work can be found in (Jelínek et al., 2008;, Novaes et al., 2006, Zhang et al., 2008). Although gold is by far the most prominent element of interest in the formation of nanowires, some recent computational studies have also involved copper nanowires (Amorim et al., 2007, Amorim et al., 2008; Sato et al., 2006a).
3. Single molecule nanowires
3.1. Experimental measurements
Building a device in which a single molecule bridges two metallic electrodes is of major interest, since one could easily tailor the nanojunction electronic properties by changing the molecule or even only a substituent in the molecule. This opens an enormous range of possibilities in the field of molecular electronics.
Most of the experimental (Cui et al., 2001; Haiss et al., 2004; Haiss et al., 2006; Huang et al., 2006; Huang et al., 2007a; Huang et al., 2007b; Li et al., 2007a; Li et al., 2006a; Xu et al., 2003a; Xu et al., 2005; Xu & Tao, 2003) and computational work (Batista et al., 2007; Hou et al., 2005; Hou et al., 2006; Kim et al., 2006b; Li et al., 2005; Li et al., 2006b; Li et al., 2006c; Paulsson et al., 2008; Perez-Jimenez, 2005; Stadler et al., 2005; Wu et al., 2005) performed with this systems deal with the measurement and/or theoretical determination of the conductance of the molecular nanojunction, particularly those in which the linker atom to the metallic electrodes is either S or N. For a recent review on some of these aspects readers can refer to (Vélez et al., 2007).
The chemical identity of the linker moiety plays a fundamental role in determining both the electrical and mechanical properties of the molecular nanojunction. Without discussion the most widely used anchoring group is thiol (Chen et al., 2006; Haiss et al., 2008; Haiss et al., 2009; Huang et al., 2006; Huang et al., 2007a; Huang et al., 2007b; Li et al., 2007b; Li et al., 2006a; Ulrich et al., 2006; Xu et al., 2003a; Xu et al., 2005;, Xu & Tao 2003), although pyridine (Xu et al., 2003a; Xu & Tao, 2003), isocyanide (Beebe et al., 2002; Kiguchi et al., 2006; Kiguchi et al., 2007; Kim et al., 2006a), selenium (Patrone et al., 2003a; Patrone et al., 2003b; Yasuda et al., 2006), amine (Chen et al., 2006; Hybertsen et al., 2008; Kamenetska et al., 2009; Kiguchi et al., 2008; Park et al., 2007; Park et al., 2009; Quek et al., 2007; Quek et al., 2009; Quinn et al., 2007; Venkataraman et al., 2006a; Venkataraman et al., 2007; Venkataraman et al., 2006b), phosphines (Kamenetska et al., 2009; Park et al., 2007) and carboxylate (Chen et al., 2006; Martín et al., 2008) have proved to provide enough binding strength to yield a stable contact. In most of these works, the metallic electrodes that the molecule bridges are made of gold, but some recent reports showed the utility of platinum electrodes for those purposes (Kiguchi et al., 2007; Kiguchi et al., 2008).
Xu et. al. reported the first electromechanical measurement of a molecular junction (Xu et al., 2003a). The authors determined simultaneously the conductance and the force under mechanical stretching for the octanodithiol (ODT) and 4,4’-bipyridine (BYP) nanojunctions. The quantum conductance for BYP resulted 40 times larger than that of ODT, while the force quantum was 0.8 ± 0.2 nN, considerably smaller than the 1.5 ± 0.2 nN determined for ODT. This last value is the same as that required to break a Au – Au bond (Rubio-Bollinger et al., 2001). Thus, the authors concluded that the breakdown of the ODT nanojunction involves a Au – Au bond rupture, whereas in the case of BYP, the lower breaking force would indicate that a Au – N bond is breaking. These results are in agreement with the notion of the Au- S bond being stronger than the Au-N bond (Stolberg et al., 1990). In a subsequent study, Huang et. al. found that the behavior of the ODT nanojunction as a function of the stretching rate is essentially identical to that of a pure gold point contacts (Huang et al., 2007a).
One useful experimental parameter to determine how strong is the molecule bonded to the metallic electrodes is the length that the junction formed by a single molecule can be stretched before it breaks. This allowed Kiguchi et. al. to establish the following order in binding energies for 1,4-disubstituted benzenes with Au and Pt electrodes: Au-NH2< Pt-NH2 ~ Au-S < Au-isoCN < Pt-isoCN ~ Pt-S (Kiguchi et al., 2006; Kiguchi et al., 2007; Kiguchi et al., 2008).
The statistical analysis of the stretching length was also used to establish the following order in the sense of increasing binding strength: Au-COOH < Au-NH2< Au-S (Chen et al., 2006).
3.2. Computational simulations
As mentioned above, a large proportion of the theoretical work on molecular nanowires involves the calculation of the conductance. Only a few of these have addressed some aspects of the thermodynamic stability of such nanocontacts.
As it respects to the mechanical properties of a molecular nanojunction, one or the first
In a different study, the stretching and breaking behavior of a benzene dithiol molecule sandwiched between two Au(111) slabs was studied using DFT calculations (Lorenz et al., 2006). It was found that breakage occurs through a dissociation of one of the Au – S bonds with a maximum force of 1.25 nN in the case when the molecule is directly attached to the surface, and of 1.9 nN when an adatom is placed between the sulphur and the gold slab.
Similar studies were carried out for nanojunctions involving 4,4’-bipyridine (Stadler et al., 2005; Vélez et al., 2005), pyrazines (Vélez et al., 2005; Zoloff Michoff et al., 2009), amines (Hybertsen et al., 2008; Kamenetska et al., 2009), and alkylphosphines (Kamenetska et al., 2009). Binding energies and rupture forces are the parameters that can be obtained from these types of computational simulations that can be related to the mechanical stability of the molecular nanojunction. It should be noticed that temperature activated processes are not considered in these calculations.
4. Long term stability of metallic monatomic nanowires
As can be gathered from the summary of the computational work performed so far with the aim of obtaining valuable information about the stability of NWs, the main challenge remains to develop models that would allow to use this valuable computational information to extrapolate to the experimental time scale, and taking thermal motion into account. In the following sections a simple kinetic model based on the
4.1. The minimum energy path and the transition state theory
A common and important problem in theoretical chemistry and solid state physics is to identify the path with the lowest energy for the reorganization of a group of atoms from a stable configuration to another. This path is referred to as the
Different methods have been developed to find the reaction path and saddle points (Michael & Michael, 2007). We focus our attention on methods that make use of two boundary conditions: the initial and final configurations for the transition. These settings should normally correspond to two local minima in the multidimensional potential energy surface. These minima may be obtained from different energy minimization techniques such as the simulated annealing, conjugate gradient, etc.
These methods require only the calculation of first derivatives of the potential energy. They generate a chain of images or replicas of the system between the initial and final configurations. All the intermediate images are simultaneously optimized in some concerted way of the potential energy surface that should be as close as possible to guarantee the convergence to the MEP. The method called
4.2. Kinetic model
As it was shown by Krüger
a bulky piece of metal of one of the tips making the junction (state P. Vélez, S. A. Dassie, E. P. M. Leiva unpublished results.
P. Vélez, S. A. Dassie, E. P. M. Leiva unpublished results.
Following the TST, we have for the forward process (wire breaking) the frequency
Let us now consider a differential elongation of the wire
We can estimate the lifetime of the wire τ* by setting in
Note that in the present formulation no assumption has been made on how the wire is elongated. We turn now to consider two different possibilities:
4.2, Results for pure metallic nanowires
In order to illustrate the method, we consider a system consisting of a Au nanowire made of a supercell containing 4 atoms which are periodically repeated in space (Au4 NW). Fig. 5 shows a scheme of the unit cell employed to simulate the Au4 NW considered here. We used 4 atoms because the elongation distances at which a NW of this size breaks are in the range between 0.11 to 0.14 nm, which is the value that Huang et al. (Huang et al., 2007a) have found experimentally (see Fig. 9a below). Although this is a rather small system, the rupture of the wire has been found to be the displacement of one of atoms perpendicular to the wire axis (Ke et al., 2007), so that the interaction of the breaking bond with the rest of the system should be minimal. Atom 1 is fixed and the length of the supercell is stretched. For a given stretching of the NW, the energy of the system can be minimized with respect to all the atomic coordinates. Let us denote with
While the unbroken state is clearly defined, some uncertainty remains concerning the broken one. With this purpose,
This configuration was then adopted to obtain the broken state for different elongations. This was achieved by compressing the system to a cell size corresponding to the desired length
Curves for the energy of the
This would yield a switching frequency between the
In Fig. 7 we can observe curves for the energies of the system between
Static rupture of the nanowire: A logarithmic plot for the static rupture of the NW calculated according to Eq. 6 is shown in Fig. 8. The lowest stretching rates employed in the experiments allow the rupture of NWs in times of the order of 0.1 s. Fig. 8 shows that in this order of times the wires should break at distances close to 0.105 nm. This is very close to the experimental value of 0.1 nm (Huang et al., 2007a) obtained at the lowest stretching rates. Thus, the present results also support the general idea that long Au – Au distances such as those found in the experiments of (Legoas et al., 2002; Ohnishi et al., 1998; Rodrigues & Ugarte, 2001a; Rodrigues & Ugarte, 2001b) at room temperature cannot occur for pure Au NWs. On the other hand, at 150 K and below, pure Au NWs can be considerably stretched beyond that point.
Rupture of the nanowire at a constant elongation rate: The dynamic rupture of a Au4 nanowire was studied according to Eq. 7. Fig. 9a shows the breaking distance as a function of ln (νe) for elongation rates between e-1 and e6 in comparison with the experimental date taken from Huang et al. (Huang et al., 2007a). It can be observed that the calculated results resemble the experimental trend in the general features.
A further point that can be analyzed through the present calculations is the lifetime of the wires as a function of the elongation rates. These results are given in Fig. 9b. There is no experimental data available for a straightforward comparison with our calculations. However, AIMD simulations (Krüger et al., 2002), and considerations based on experiments (Huang et al., 2006; Huang et al., 2007a; Li et al., 2006a) indicate that the rupture of a nanocontact made of an alkanethiol and Au contacts should break at a Au – Au bond, so that comparison between the present results may be made with the experiment of Huang et al. (Huang et al., 2007a), who have studied the rupture of single molecule junctions involving Au contacts and ODT. The experimental data of Huang et al. is included in Fig. 9b, where it is found that the calculated lifetimes closer resemble those from the experiment; specially taking into account that no fitting attempt was made seeking for agreement.
4.2. Results for contaminated metallic nanowires
Fig. 10 shows a scheme of the unit cell employed to simulate the pure (Au-p) and contaminated Au NWs considered here. The light weight elements taken into account as contaminants are a H atom (Au-H) and a C atom (Au-C). The grey circles represent Au atoms, which remain fixed at their positions during the relaxation processes. The latter consists in a local energy minimization procedure by means of the conjugate gradient method or a search of a MEP by means of the NEB method.
In the case of contaminated NWs the circle marked with an X represents the location the contaminant atom. This figure also shows the definition of the α1, α2 and α3 bond angles, determined by the atoms relevant for the analysis of the rupture of the NW.
The length of the NW, L, is defined here as the distance between the atoms Au1 and Au5. We also define an average Au – Au separation, as
The atomic impurity was located between the atoms Au1 and Au2. This choice was made because in the literature we found first principles calculations, similar to those performed here, where the H and C were positioned at a similar place, as well as between the Au2 and Au3 atoms, with similar results (Novaes et al., 2003; Skorodumova et al., 2007). As it will be found below, the present results agree with those where the impurity was located at another sites of the chain.
Structure and energetics of pure and contaminated Au NWs. For all the systems we shall refer to the equilibrium state as that where the derivative of the energy with respect to the elongation is equal to zero (ie. the external force, or stress, acting on the system is null). The rupture force will be considered to be the value of the force Fz where it presents a maximum at long elongations, being the force defined in Eq. 8 (da Silva et al., 2004; Jelínek et al., 2008; Novaes et al., 2003; Novaes et al., 2006; Rubio-Bollinger et al., 2001; Vélez et al., 2005; Vélez et al., 2008). Accordingly, we refer in the following to the “at rupture” state as that where the coordinates of the atoms are such that
Considering the length difference between the “at rupture” and equilibrium states of the different systems, it is found that the stretching lengths of the Au-p and Au-C systems are 1.9 Å and 1.6 Å respectively, while the elongation of the Au-H system is considerably larger (2.6 Å). This fact bears direct consequences for the force constant kz, as will be seen later on. Inspecting Table 1 it becomes clear that the equilibrium geometries of the contaminated
|Equilibrium geometries||„At rupture“ geometries|
|Le or L* / Å||9.80||9.60||10.8||11.7||12.2||12.4|
|d (1,2) / Å||2.63||2.76||3.78||2.85||3.58||3.87|
|d (2,3) / Å||2.64||2.62||2.59||2.95||2.75||2.67|
|d (3,4) / Å||2.63||2.67||2.66||3.05||3.16||3.15|
|d (4,5) / Å||2.63||2.61||2.62||2.85||2.71||2.71|
|α1 / °||139.5||149.6||161.8||179.3||178.6||178.5|
|α2 / °||125.2||108.6||116.5||179.7||177.0||176.5|
|α3 / °||139.7||157.4||136.0||179.9||178.5||178.2|
|αX / °||---||102.5||168.5||---||179.2||179.5|
systems are very different from each other. Close to equilibrium, the presence of the H atom changes only slightly the d (1,2) bond distance, while the carbon atom inserts itself almost completely in the middle of this bond. This fact can be appreciated in the characteristics of the equilibrium configuration (see αX bond angle and d (1,2) bond distance).
At the situation of the rupture, all three systems present a linear configuration and the distance of the Au3 – Au4 bond where it occurs is very similar (~3.1 Å). The contaminated systems arrive to the point of rupture with large Au – Au distances at the Au1 – Au2 bond; 3.58 Å for the Au-H system and 3.87 Å for the Au-C system. These figures indicate that our results are in a very good agreement with the Au – Au separation distances found experimentally (Kizuka, 2008; Kondo & Takayanagi, 2000; Legoas et al., 2002; Ohnishi et al., 1998; Rodrigues et al., 2000; Yanson et al., 1998) and with the first-principles calculations from other research groups (Novaes et al., 2003; Skorodumova et al., 2007).
Up to now, we have only explored the possibility of the incorporation of a single atomic impurity. In relation to this, it must be recognized that the extremely large Au – Au distances of 4 – 5 Å (Legoas et al., 2002), as well as consecutive distances of 3.5 – 4 Å (Kondo & Takayanagi, 2000; Ohnishi et al., 1998; Rodrigues & Ugarte, 2001a; Rodrigues et al., 2000) cannot be reproduced by our calculations. To tackle this point, more impurities should be considered, and probably other molecular species as proposed by other groups that performed first-principles calculations (Bahn et al., 2002; Galvão et al., 2004; Legoas et al., 2002; Novaes et al., 2003; Skorodumova & Simak, 2003; Skorodumova & Simak, 2004). However, the present approach is not devoted to predict all the Au – Au distances observed, but to understand the effect that an atomic impurity produces on a monatomic Au NW from energetic, geometrical, electronic and kinetic viewpoints.
Analysis of the time-stability of pure and contaminated NWs. In the following the kinetic aspects of the rupture process for the Au-p, Au-H and Au-C systems are considered. Fig. 12a shows the energy curves for the unbroken (u), broken (b) and activated (#) states as a function of the elongation (Δz) for Au-H. For Au-p the respective curves are similar to those shown in Fig. 6 for the 4-atom gold nanowire. The results obtained for Au-C have qualitatively the same features as those obtained for Au-p.
In order to find candidate configurations for the broken (final in TST jargon) state at each elongation, molecular dynamics runs were performed at an elongation corresponding to the “at-rupture” configuration. This led to structures that looked like those illustrated for Au-H for the broken state configurations in Fig. 14 below.
We will perform a more detailed discussion of the reaction path for the rupture of the NWs below. We first consider the behaviour of the forces along the stretching procedure, since they are closely related to the energy curves of the unbroken state. In fact, the energy curves for the unbroken states in Fig. 12a may be used to calculate the longitudinal force Fz acting on this system according to Eq. 8. In Fig. 6, it was found that the stability limit of a monoatomic Au NW is reached at a force of 1.45 nN, which is close to the experimental value of 1.5 ± 0.3 nN (Kizuka, 2008; Rubio et al., 1996; Rubio-Bollinger et al., 2001). Fig. 12b shows Fz for the Au-H as a function of the elongation Δz. The maximum force Fz max is also reported there. Table 2 compiles relevant information for this and the other systems under consideration in this study (Au-p, Au-C). For the Au-p system, the maximum force is 1.57 nN, also in agreement with the experiments and other theoretical values (da Silva et al., 2004; Rubio-Bollinger et al., 2001; Vélez et al., 2008). In the case of the contaminated systems, this value is somewhat lower (1.19 nN for Au-H and 1.14 nN for Au-C). First principles results from literature show the same trend (Novaes et al., 2003; Skorodumova et al., 2007).
|Fzmax, nN||kz, N/m||ΔE‡(Δz), eV(Å)|
|Au-p||1.57||9.1||0.99 (0.0)||0.22 (1.8)|
|Au-H||1.19||5.0||1.33 (0.0)||1.02 (2.7)|
|Au-C||1.14||8.2||1.12 (0.0)||1.10 (1.8)|
In their studies on the mechanic properties of monatomic Au NWs, (Rubio-Bollinger et al., 2001) found that these chains are five times harder than the massive electrodes. They evaluated from the experimental results the slopes of the force curves in the last stage of the elastic deformation, before the rupture of the NW, from a set of 200 experiments. The average value of the force constant for an average chain length was 8 N/m. The present calculations show that the behaviour of the force curves of the Au-p, Au-H and Au-C systems is quite elastic, as can be inferred from the linear fit of Fz between zero and the maximum force, shown in Fig. 12b for Au-H as a broken line. In the elastic deformation region, the force Fz may be written as:
where kz is the force constant of the system. The value of kz found from the present calculations for Au-p, 9.1 N/m, is in perfect agreement with the experimental value reported in references (Rubio-Bollinger et al., 2001; Xu et al., 2003a). The contaminated systems present a remarkably different behaviour when compared between each other. The Au-C system has kz =8.1 N/m, a very similar value to that of Au-p, but 1.6 larger than the value of this property for Au-H (kz=5.0 N/m). These are interesting predictions, since up to date no measurements have been performed comparing pure and contaminated systems. A suitable fitting of the experimental data of this property could help to shed light on the type of impurity present when a NW exhibits large Au – Au separation distances.
Fig. 13 shows the energy of the system, E, as a function of the normalized reaction coordinate (NRC) for the system Au-H for some sample elongations. The corresponding configurations are presented in Fig. 14.
Fig. 13 shows that the Au-H system presents an energy curve along the minimum energy path that is more complex than those for Au-p and Au-C (not shown here). The distinctive feature for of Au-H is that at short elongations the energy curves present a minimum. This behavior is related to the incorporation of the hydrogen atom into the wire that takes place as the elongation proceeds. However, for the Au-C system, the C atom is incorporated into the wire since the beginning of the elongation. The minimum in the minimum energy path profile of the Au-H system disappears gradually with increasing elongation of the NW, remaining a shoulder at long elongations (see Fig. 13b).
The images of monatomic Au chains showing long separation distances (3.6 – 4 Å) were obtained by TEM and HRTEM using electron beam lithography for the fabrication of the NWs (Kizuka, 2008; Kondo & Takayanagi, 2000; Legoas et al., 2002; Ohnishi et al., 1998; Rodrigues & Ugarte, 2001a; Rodrigues et al., 2000). Under these experimental conditions, the elongation rate is not controlled and in principle not known. However, an estimation can be made looking at the pictures provided in some of these publications, as the one illustrated in
Fig. 2 (Rodrigues & Ugarte, 2001a), and is found to be very slow, of the order of 0.1 nm/s in the final stage, allowing for a complete equilibration of the system at the atomic scale all along the elongation process. Thus, the rupture of the NWs in these experiments takes place in the timescale of a second, so that it can be inferred that the activation barriers must be quite high. In the static limit, the lifetime τ* of a NW can be estimated from Eq 6. In this equation, we have used νu(Δz) = νu = 3 x 1012 Hz and T = 300 K. Similar values were used in references (Bürki et al., 2005; Rubio et al., 1996; Todorov et al., 2001; Vélez et al., 2008). Fig. 15 shows a plot of the decimal logarithm of the lifetime τ* of the NW, as a function of the elongation force Fz for the Au-p, Au-H and Au-C systems.
It becomes evident from this figure that the contaminated NWs live considerably longer than Au-p NWs for all elongations. It is found that pure Au NW become unstable in the experimental time scale from an elongation which corresponds to dAu-Au = 2.74 Å / atom on. In fact, for larger Fz (dAu-Au), pure NWs should live less than 0.1 s. The remarkable behaviour of the Au-H system is due to the shape of the activation energy, discussed above in Fig. 13. Fig. 15 also supports the idea that impurities are responsible for the large Au – Au distances observed experimentally. These impurities, as stated above, modify geometrically the structure of the NWs, giving place to the occurrence of the anomalous large Au – Au distances. Furthermore, they modify the NW chemically by charge transfer and rearrangement, which in turn changes the potential energy surfaces so that for each elongation the contaminated NWs present higher activation barriers for the rupture than the pure NWs. Fig. 15 makes also plausible that stretched (contaminated) NWs may have lifetimes of the order of the second, sometimes even of the order of minutes (Ohnishi et al., 1998).
5. Long term stability of molecular nanowires. Effect of substituents on the Au – N nanojunction.
The formalism proposed in the previous sections could, in principle, be applied to study the mechanical and kinetic stability of single molecule nanowires. To the best of our knowledge, no such study has been yet attempted. One possible approximation to the long term stability problem is to estimate the kinetic barrier for the rupture of the nanowire by means of the binding energy of the system (Eb).
The mechanical properties of ortho substituted pyrazines bonded to planar gold clusters of 6 and 7 atoms have been studied using the same methodology described for the monatomic metallic nanowires. In Table 3, we summarize the values obtained for the binding energies (Eb) and rupture forces (Frup) for the substituted pyrazines bonded to the 6-atom (Au6) and the 7-atom (Au6+1) gold clusters.
The systems studied are illustrated in Fig. 16. These systems are taken as a model for the last stage in the stretching of a Au – bipyridine nanojunction, and allow an assessment of the effect of a substituent in the molecule on the stability of the nanojunction.
The Eb values summarized in Table 3 were obtained as the energy difference between the minimum energy structure for each system and the final broken structure after stretching the nanojunction. In all cases the rupture occurred at the Au – N bond. Frup values were obtained using Eq. 8.
|Au6 + X-pyrazine||Au6+1 + X-pyrazine|
|X||σ||Eb, eV||Frup, nN||Eb, eV||Frup, nN|
Using the Eb values as a lower limit for the activation energy to break the nanojunction, and using Eq. 6 with νu = 1012 Hz (a typical value for these systems as shown above), time constant values, τ*, can be calculated for the different nanojunctions. The values for τ so obtained represent a lower limit for the lifetime of the nanojunctions. Interestingly, there is a an excellent correlation between the logarithmic lifetime and a pure empirical parameter describing the electronic nature of the substituent such as Hammett’s σ (Hansch et al., 1991), as illustrated in Fig. 17a.
Fig. 17b shows that a good correlation is also observed with respect to the calculated rupture force for the nanojunction. This is interesting since Frup values can be obtained experimentally.
Note that the electronic properties of the substituent has a marked effect on the lifetime of the nanojunction, which ranges from 10-6 seconds for X = NO2 to about 103 seconds for NH2. On the other hand, only a slight effect of the substituent was found on the experimentally measured conductance of a structurally similar system (Venkataraman et al., 2007).
When the molecule binds to a less coordinated gold atom, such as in Au6+1 – X-pyrazine systems, the strength of the nanojunction increases, as reflected by the higher values obtained for both Eb and Frup. This also causes an increase in the lifetime of the molecular junction, as illustrated in Fig. 17.
6. Conclusions and perspectives
A comprehensive revision of the most recent advances with respect to the experimental techniques and computational simulations focused on the stability and mechanical properties of monatomic metallic and single molecule nanowires is presented.
We have established a simple model based on the Transition State Theory and using the Minimum Energy Path to study the long-term stability of nanowires in a time scale corresponding to that of the experimental observations. The utility of this model has been demonstrated for the quantitative evaluation of the effect of impurities in gold nanowires, as well as for a qualitative assessment of the effect exerted by a change in the electronic properties of a molecule in the temporal stability of a metal – molecule junction.
Some of the future work within this topic includes the analysis of other systems, including different metals and / or other chemical linkers for molecular nanowires. It would also be of great interest to evaluate, if there is any, the correlation between the stability of the nanowire and its electronic properties, such as conductance.
- Times Online September 19, 2007, http://technology.timesonline.co.uk/tol/news/tech_and_web/article2489053.ece
- P. Vélez, S. A. Dassie, E. P. M. Leiva unpublished results.