Atomistic simulations of competing influences on electron transport across metal nanocontacts

Résumé

The quest for ever smaller transistors is an ongoing endeavour which today mainly focuses on simultaneously exploiting the electron’s charge and spin, in an attempt to maximise information processing power via concrete realisations of spintronics [1]. However, the success of this venture is conditioned by the fact that, at the nanoscale, every atom and spin now counts. For this reason, understanding the physical behaviour of materials, right down to the nanoscale, is of significant technological importance, and the pursuit of such understanding, continues to pose new experimental and theoretical challenges. Already in 1992, Professor Uzi Landman, who is famous for pioneering the field of emergent properties of materials at the nanoscale, showed through careful analysis [2], how measurable physical and chemical properties of various materials scale with size. Interestingly, he found that certain properties do not scale in a predictable manner all the way down to the nanometer-scale. That is, at the nanoscale, certain properties of materials depart from the usual scaling laws that apply to the macroscopic systems. Landman et al. [3] have continued their research into a growing number of nanoscale systems that exhibited emergent properties, including more recently, exotic systems that are related to the burgeoning field of molecular machines [4,5]. In fact, the 2016 Nobel Prize in Chemistry was awarded to Jean-Pierre Sauvage, Sir J. Fraser Stoddart, and Bernard L. Feringa “for the design and synthesis of molecular machines”. From a practical standpoint, the non-scalability of measurable properties of nanoscopic materials, poses serious challenges to the realisation of nanoscale devices for technological applications. Within this broader context, the research presented in this thesis is related to some specific questions that arise in connection with nanoscale electrical contacts between metals. Recent, groundbreaking work has shown that thermal transport is nonscalable at the atomic level: atomic-sized Au contacts were shown to transport heat in discrete quantised packets [6]. Previously, it had been known since the 1990s that charge transport is quantised in atomic-sized contacts made of metals such as Au [7]. Initially, the quantised charge transport was attributed to conduction through discrete energy levels, i.e., waveguides, resulting from lateral confinement of electrons in the atomically narrow contacts [8]. However, the very first classical molecular dynamics (CMD) simulations of atomic-sized metallic contact formation, performed by Landman et al. in 1990 [9], suggested another explanation for the quantisation of charge transport in metallic nanocontacts. The grainy nature and geometry of atomic-sized contacts, in combination with the chemical valency of individual atoms within such contacts, should determine charge transport, when only a single or few atoms occupy the minimum cross-section of the contacts [10]. As experiments increasingly provide more details, and are able to quantify more properties, theoretical models must necessarily also become more accurate. Models must now take into account many more subtle quantum mechanical effects, such as, the effects of lattice motion on magnetism, and higher order relativistic corrections. The development of one such model, which can be applied not only to nanocontacts, but also to other systems, is the focus of this work. Despite all the research devoted to the study of metallic nanocontacts during the last decade (see Refs. [11,17] and references therein), there are still unexplained phenomena and new emergent properties observed for these contacts. An example from the above list is the differences in the jump to contact behaviour of Au, Ag and Cu nanocontacts, not fully addressed until this work. In this thesis, the influence of relativistic effects is shown to provide the explanation (see Chapter 5, Section 5.2). Another example is the role of non-collinear magnetism in the unexpected low-conductance features of the experimental conductance histograms of ferromagnetic nanocontacts made of Fe or Ni. Are these features perhaps the result of ballistic magnetoresistance due to magnetic domain walls in the contacts [18]? In view of the last two examples, which are described in more detail and posed as research questions in the next section, a method is needed that incorporates scalar- and vector-relativistic corrections in the standard computational toolbox that currently exists for the study of atomic-sized point contacts, namely, that of CMD simulations of the lattice dynamics [19] and DFT quantum transport calculations [20]. In this work, a computationally efficient method is developed to explore the relative importance of relativistic effects on the emergent properties of representative noble-metal and ferromagnetic transition metal nanocontacts in low-temperature experiments. The emergent properties include the unusually large jumps to contact observed for Au vs Ag or Cu nanocontacts, the anomalous low-conductance peak structure in experimental conductance histograms of Ni nanocontacts, or the unusually high position of the first conductance peak of Fe nanocontacts in experimental histograms. More specifically, this thesis seeks to address two research questions that are of fundamental importance to the understanding of bonding and electrical resistance at the atomic scale in transition metal nanocontacts. The first question concerns the bonding strength in noble-metal nanocontacts, made of Cu, Ag or Au, respectively. Since these elements occur within the same group (11) of the periodic table, one would expect their bonding strengths to be comparable to one another. What then accounts for the much stronger bonding strength between nanoscopic surfaces made of Au –observed as a much larger jump to contact in conductance– than either Ag or Cu in STM/MCBJ experiments, as reported and partially addressed in Refs. [21–24]? Can scalar-relativistic effects, which are extremely important in 5d metals such as Au, as most recently claimed in Ref. [17], explain this difference? In this thesis, pseudopotential and all-electron plane-wave DFT calculations are used to compare the interaction energies of infinite one-dimensional monatomic wires made of Au or Ag, as a function of interatomic separation [25–27]. Scalar-relativistic effects, spin-orbit coupling [27] and van der Waals forces [28,29], are explicitly included and excluded in order to evaluate the different contributions and highlight the central importance of (scalar) relativistic effects in gold. The second question concerns the competing roles of the directionality of bonding between atoms, on one hand, and (non-collinear) magnetism, on the other, in ferromagnetic 3d-metal nanocontacts. Do vector-relativistic effects, i.e., spin-orbit coupling, and the non-collinear magnetism it gives rise to in the nanocontacts, affect the latter’s structural evolution and measured conductance? Or, is the distribution of atomic configurations, giving rise to peaks such as those of Fig 1.2 b), governed by the extent of covalent bonding within the materials? Clearly, it is pertinent to understand the relative importance of these two competing effects, particularly when the nanocontacts are about to form, or break. In this thesis, I will apply my model to Fe (a BCC metal) and Ni (an FCC metal), in order to shed more light on this second question. In the case of Fe nanocontacts, a discrepancy has been observed between the first-conductance peaks of theoretical and experimental conductance histograms constructed from contact-rupture trials at 4.2 K in Ref. [30]. Is this discrepancy, as the authors contend, a fundamental limitation of the interatomic potential, of the EAM type [31], used to generate last-contact atomic structures in their CMD simulations? This conclusion was reached by the authors after trying several different EAM potentials. These potentials treat the bonding in metals as isotropic, an obvious shortcoming in the case of BCC metals, since they exhibit much greater covalent character than FCC metals. The relatively greater covalency in BCC metals is exemplified by the 4 fewer first-nearest neighbors in a perfect BCC vs FCC lattice. As an alternative theory, perhaps the formation of magnetic domain walls (DWs) at the moment the iron contacts are about to rupture, can explain the above discrepancy? DWs are known to affect conductance in ferromagnetic nanocontacts [16,18]. Similarly, magnetic DWs have been proposed [32] as a possible explanation for the anomalous peak structure seen in conductance histograms recorded for Ni nanocontacts in STM/MCBJ experiments [32–34]. Those histograms exhibit varying peak structure at first or last contact, when as few as one atom bridges the electrodes comprising the nanocontact. Most often [30,35–38], just a single broad peak centred at a conductance of ~1.5 is obtained. However, experiments have been performed in which sub-peak structure is revealed to be concealed by the single broad peak: a double-peak with a first maximum at at ~1.2, and another at ~1.5 [32–34]. Once more, what is behind this phenomenon, magnetic DWs pinned at the constriction at the moment of rupture or contact formation [39]? The peak at 1.2 might correspond to a situation where an abrupt DW is present at the constriction, while when no DW is present, the other peak occurs. On the other hand, perhaps two different preferential last-contact structures, vertical dimers (inset of Fig. 1.1 b)) and monomers (inset of Fig. 1.1 a)), very commonly generated by EAM potentials in stretching simulations, along two different crystallographic orientations of FCC Ni [32,40–42], might correspond to the two different low-conductance peaks? In this thesis, the second research question is explored primarily by combining, for the first time, CMD and SLD simulations, to study ferromagnetic nanocontacts made of iron and nickel. SLD is used in combination with the best available EAM potentials for these metals [43,44] to see whether or not the presence of spin-lattice coupling can affect the type of atomic arrangements that arise at the moment the contacts are about to rupture or form. As an alternative possibility, a very recent modified embedded-atom method (MEAM) interatomic potential [45], which is, as the name suggests, a modified version of the EAM potential in which the bonding has directional character, is used to explore the type of first- and last-contact structures these metals adopt in simulations. Finally, to permit direct comparisons with the experiments, DFT electronic transport calculations, up to the vector-relativistic level of sophistication (including non-collinear magnetism), are employed to calculate the conductance of snapshots extracted from CMD and SLD simulations. Disentangling the roles of competing geometric, electronic and magnetic effects in electronic transport through atomic-sized metallic nanocontacts is both important from a fundamental point of view, since it teaches us about the importance of these phenomena in the limit of a single atom and in bonding between a few atoms. In technological applications, it is crucial to understand these effects on spin-polarised transport, i.e., in spintronics [1], one of the most active and promising research fields in quantum computing. Relativistic effects play a central role in heavy transition metal elements such as Au, by, for example, leading to smaller than expected lattice constants for these metals, as a result of the contraction of their valence s orbitals, among other phenomena [66]. In low-dimensional systems such as nanocontacts, relativistic effects are expected to lead to even more exotic phenomena than in the bulk metals, such as the formation of suspended monatomic chains, several atoms long, when Au contacts are ruptured [67]. Studying how relativity affects the electronic transport properties of Au nanocontacts therefore leads to a better understanding of bonding between the atoms in these low-dimensional systems. Ferromagnetic nanocontacts have never been modelled by SLD with spin-orbit coupling before, which provides a unique opportunity to study how (non-collinear) magnetism and atomic structure interact when the nanocontacts evolve dynamically under cyclic loading. Two important challenges, therefore, arise in the modelling undertaken in this work: the coupling between the lattice and the atomic spins, which implies making use of combined CMD and spin dynamics, or spin-lattice dynamics, and the inclusion of spin-orbit coupling and non-collinear magnetism in DFT transport calculations. As a result, modifications of the source code of widely used simulation software, e.g., the Large-scale atomic/molecular massively parallel simulator (LAMMPS) [50] and spin-lattice dynamics (SPILADY), as well as ANT.Gaussian [68], or of their parameters, have been developed as part of the work presented in this thesis. These newly-developed tools, far from being applicable to only the systems studied here, can be applied to other exciting low-dimensional materials of current interest, such as ferromagnetic thin films and nanowires. These materials are promising candidates in non-volatile memory applications [69]. Hence, the tools developed in this work can be extended to study, for example, the stability and dynamics of mobile Skyrmions [70] on thin films, and transverse domain walls [71] in ferromagnetic nanowires, in the presence of defects and temperature shocks and gradients. In summary, the new tools open a whole new avenue of research into low-dimensional systems where magnetic and structural degrees of freedom are intimately coupled. In order to tackle the study of both ferromagnetic and non-magnetic metallic nanocontacts, the following models have been applied and extended: • The spin-lattice dynamics (SPILADY) developed by Ma et al. [48] has been extended to include magnetic anisotropy and non-collinear magnetism for nanostructures. This model has been parameterised for Fe and Ni. • Spin-orbit coupling has been implemented in the electronic transport code ANT.Gaussian. The method has been validated by comparison to vector-relativistic self-consistent calculations done in OpenMX. These calculations and implementations required making use of different simulation packages such as ANT.Gaussian, OpenMX, CRYSTAL14, CASTEP and Wien2K for DFT, as well as LAMMPS and SPILADY for classical molecular dynamics and spin-lattice dynamics. The extended methods have been applied to study metallic nanocontacts, ultimately providing new insight into the two research questions that were posed. Concerning the first research question, about the much larger jump to contact measured in the conductance for Au, than in either Ag or Cu, we have seen that: • A study of the stability of atomic contacts in the three metals, Au, Ag and Cu, via a combination of DFT and CMD, reveals that electronic transport across these structures depends crucially on the number of first neighbours. • Relativistic effects explain the enhanced bonding in Au compared to Ag or Cu and consequently the experimental observations in jump to contact behaviour, which cannot be explained by any other proposed factors such as van der Waal’s forces, spin-orbit coupling, or elastic constants along different crystallographic orientations. Concerning the second research question regarding the extent to which spin-orbit coupling or covalent bonding may explain the anomalous peaks observed for iron and nickel, we have seen that: • The discrepancy between experimental histograms of conductance for Fe nanocontacts and previous calculations, has been studied in detail. After exploring different contributions, from magnetic effects to electronic structure, the calculations carried out in this work indicate that the difference is related to the BCC structure and the formation of very stable contacts before rupture that are several atoms across. These structures are produced during rupture due to a transformation from (001)-oriented BCC layers to (110)-oriented ones perpendicular to the direction of stretching. Such structures give rise to a conductance value, obtained from DFT electronic transport calculations of ~2, in good agreement with experimental measurements, unlike previous calculations. • Experimental Ni histograms of conductance also exhibit some unexplained behaviour. They can vary from one experiment to another, exhibiting in most cases a broad peak at 1.5, while in others, which are less frequent, two peaks are observed. The presence of domain walls at the nanocontact influencing the conductance in this material had been proposed as a possible explanation for this behaviour. However, our simulations, using spin-lattice dynamics, which have been applied to these systems for the first time, show that, like in previous DFT calculations, domain walls make a very small contribution to the conductance in these systems when no external magnetic field is applied. On the other hand, our classical molecular dynamics calculations show that there is a difference in the most stable atomic structure before rupture depending on the orientation of the lattice, in particular for (111) and (001). Moreover, the spin-lattice dynamics also show a stronger influence of the presence of domain walls in the (111) than (001) orientation, although we should note that these calculations have been set up so that MR is maximised, in order to determine the maximum possible effect of DWMR. Therefore, we propose that a combination of the lattice orientation together with the influence of domain walls, to a lesser degree, could explain the variability observed experimentally, especially when (111)-oriented structures occur, since cyclic loading in these structures lead to elongated nanocontacts and the formation of domain walls are also most favoured in them. Further studies will have to be undertaken to confirm this hypothesis. Besides the results obtained for nanocontacts, the models developed in this thesis can be used in many other applications to study phenomena such as defects in magnetic materials, magnetic surface effects or Skyrmions, interaction among magnetised islands (quantum dots) on non-magnetic surfaces, among others.