Resumen de "Ab initio" study of low-dimensional metallic systems
Summary of the PhD thesis ¿Ab initio calculations of low-dimensional metallic systems¿ In the present Doctoral Thesis we gather all the research performed by the graduate student Jorge Botana Alcalde. Broadly speaking, the work consisted in the study of the structural, electrical and magnetic properties of a spread of low-dimensional systems, made up with transition metals. We can divide said systems in two categories: ultrathin film of iron and chromium, where the two iron layers are sandwiched around a central chromium layer, and atomic clusters, either made of gold or of an alloy of bismuth and manganese.
This work was done using ab initio techniques, i.e. from first principles. In our systems that means to solve the non-relativistic Schrödinger equation for a many-electron system. Our approach to such a complex problem was by means of the density functional theory (DFT), which transforms the many-body problem, considering each electron interacting with every one of the other electrons, to a single-body problem, where each electron interacts only with an effective potential created by the electron density. The implementation of this approach for the calculation of our systems was performed using the deMon (density of Montreal) DFT software.
Trilayers of Fen/Cr3/Fen (n = 1¿ 7) The ultrathin multilayers are among the most widely used nanostructed systems within the current computational technology. Their properties had already been subject of extensive research by the time we had started our own work on them.
This research had chiefly explored to great depth the dependence of the magnetic properties of multilayers with the spacer layer thickness, due to, as we have seen, the modulability of the GMR effect depending on said magnitude. In the meanwhile, the effects of the FM layer thickness had received little attention. This led us to think it was a topic with great promise, as the FM layer thickness affects the properties of the multilayers, as found by Bloemen and collaborators.
The magnetic multilayers are often studied theoretically through modelization. They are parameter-adjusted models that deal with electronic and magnetic transport properties, readily measured in experiments, and come from the continuity equations like the Boltzmann equation. Nevertheless, to understand why this transport (macroscopic property) happens, we first need to determine how the magnetic moments of the atoms of the trilayer couple with each other, we need to know their electronic and magnetic properties (microscopic properties). This is what we attempted to achieve with through our ab initio calculations.
Our goal was to find the magnetic structure of these trilayers from ab initio calculations and to discuss the electronic distribution, looking for the anomalies and dependencies that might happen when increasing the FM layers thickness. In order to optimize the simplification of the problem, we decided to use Fe/Cr supercells, first studied by Okuno and collaborators, and then took the simplest supercell system, which consists in two FM layers (made of Fe) separated by a spacer (made of Cr): a magnetic trilayer. While it would seem the best choice for a spacer layer is a non-magnetic material, this needs not to be the case. Cr is often selected to go with Fe because it couples AFM with it (in addition to Cr monolayers being coupled AFM with each other), which increases the energy difference between the parallel and antiparallel states of the FM layers. For our calculations, we fixed the thickness of the Cr slab to 3 MLs. The distance between planes is a/2, being `a¿ the lattice parameter. The thickness of the Fe slab will vary between 1 and 6 MLs. All the atoms, either Fe or Cr are considered into a bcc lattice with a lattice parameter of a= 2.88Å.
We found the magnetic structure, layer by layer, of our Fe/Cr/Fe trilayers. The results we obtained improved and added new data to previously published results for these systems. The MMA of the Fe atoms are parallel, and their values are close to the bulk value, being higher for the surface Fe ML and lower for the interface Fe ML. The MMA of the interface Cr atoms are slightly smaller than the bulk value, and almost zero for the middle Cr ML. The Cr MLs couple AFM with each other and with the Fe ones. This related with previous publications about Cr spin-waves, only that those Cr slabs were much thicker than ours. Hence, the Cr MMs follow the profile we would expect from a commensurate SDW, as it would corresponds to the position of our Cr slab in the Cr phase diagram, except that SDW behaviour is not well defined for such a thin slab. The ¿T of the unit cell was found to follow a linear law.
We also studied the charge distribution through MPA. The most notable feature was that the Fe interface MLs are depleted of electrons, which accumulate in the Cr central ML. There are small population deviations with respect to the isolated atom state in the rest of the MLs, depending on the thickness of the Fe slab. Having all this in account, we found that these deviations fit to a Friedel-like oscillation.Considering this, it was possible to obtain a rough approximation for the m* in the systems 232, 333, 434, 535 and 636. m* ranged from 0.1 to 0.5 times me, and decreasing with the iron thickness layer. The shape of the oscillation indicated that, in terms of charge distribution, Cr layers could be considered to be acting as a perturbation that will accumulate electrons, perturbation whose influence decays inversely with distance.
The density of states for the atoms of the unit cell, each representing a different ML, was calculated as well. We found that the MLs that rule the behaviour of the properties based on the DOS are the Fe surface, the Fe interface and the Cr interface. The Fe surface ML has always a part of the total resistivity that is due to the DOS(EF) much larger for spin down electrons than for spin up ones. Finally, we have also managed to draw a qualitative image of one of the factors that contributes to the spin-dependent resistivity in the trilayer interfaces of these systems. We found a short range oscillation of this factor, which can be important in the systems 131 and 333 and 636.
Very small gold clusters.
Within the current trends of the research in nanoscience, the development and study of structures with a low number of atoms (clusters and nanoparticles) has become a topic of major interest, where different fields of knowledge (Physics, Chemistry, Biomedicine and Engineering) converge. The noble metals: Cu, Ag and Au in particular are the focus of much of this research. The Au clusters, particularly, show a widespread of very interesting properties that are being predicted: interaction and stabilization of DNA, relatively high magnetic moments, structure-dependent adsorption of amino-acids, role in organic catalysis, properties of molecules where they interact with sulphur. Most of these properties of Au clusters come from the low-dimensionality structures they conform, in contrast with Ag and Cu. It could be thought that Au, Ag and Cu, being isoelectronic, would play an interchangeable role in molecules or atomic aggregates. This is often the truth, especially when it comes to chemical properties. The clusters of these three elements adopt planar structures from one to eight atom size. But, while Ag and Cu 8-atomclusters have been shown to adopt the geometry of a distorted bi-capped octahedron with symmetry D2d, ab initio calculations have fairly established that the 8-atom gold cluster is shaped like a 4-point star, a tetra-capped square with symmetry D4h. This ground state was found in calculations with both pseudopotentials and perturbative methods.
Au clusters keep their 2D character up to 12 atoms: in this cluster, the lowest energy structure is already tridimensional. The reason for this late 2D-3D transition in the Au small clusters has attracted considerable attention.
Before our work, this had been found to happen due to strong d-d orbitals overlap in the 2D structures, which makes them energetically competitive with respect to the 3D ones. The Ag and Cu clusters are less electronically dense and there will not be as much d-d overlap, hence the 2D structures will not be energetically favoured. Using a different approach, we show that the d-d overlap actually lowers the energy for 3D Au clusters, and the exchange¿correlation (XC) and the electron-electron (e-e) repulsion favours 2D geometries.
Ab initio calculations using the Density Functional Theory (DFT) has been throughout the main tool and playground to explore the geometry of these very small clusters. How can it help to also understand the reasons behind them adopting one structure or another? To understand what favours one certain geometry over others we have to go beyond the comparative analysis of the different isomer energies. We need to study the properties of their electronic structure, such as the shape of the orbitals and the electron density and localization on different structures for the 8-atom neutral clusters of Au and Ag. From this comparison, we will try to deduce why Au clusters tend to arrange in 2D structures.
Is there a relationship between the structural deformations of the 3D clusters with the fact that 2D clusters are energetically much more favoured in Au than in Ag and Cu? From our own and others¿ works, we know that the minimum energy structures for Au and Ag have a very small difference when considering the average distance between nearest neighbours (within 1%), and these distances are sensibly higher for them than for Cu. This reduced size is well known to be due to relativistic effects. This relativistic shortening of the size of the Au clusters increases the classical Coulomb repulsion among the Au nuclei and among the electrons of the cluster, hence favouring structures where the average distance between atoms is larger. The two-dimensional structures have a larger average distance between atoms even if the average distance between first neighbours is smaller, because the coordination number is lower. It has been observed for larger Au clusters (from 20 to 50 atoms) that while they are 3D, their lowest energy structure is not tight-packed like, but instead, they tend to form cage-like structures. We have found that this is the case too even for clusters as small as 8 atoms, considering only their 3D structures. The 3D clusters deform in a way that reduces the coordination number, and create ¿cavities¿ in their inside. The apparition of these cavities causes the electron localization function value to drop in them, hence making the electrons to be more localized in the 3D clusters than in the 2D ones. This higher delocalization of the 2D structures will make the XC energy factor to be favourable to them (which, as we have seen, is the decisive energetic factor), stabilizing them as the lowest energy structure. Our findings conflict with past works on this topic that shift the control of the dimensionality to the kinetic energy term. In their studies they find that the kinetic energy favours 2D structures because of the importance of d-d interaction. We have shown that this d-d overlapping is actually more important in the 3D clusters, hence the kinetic energy will be lowered for them. Au cluster had been studied in comparison with the 8-atom Cu one. While it had been found that the XC energy controls the dimensionality of the Cu clusters, they did not retrieve this behaviour for the Au case. We have shown that for Au, and Ag too, the XC energy is still the energy term that controls the cluster dimensionality.
While 8 atom gold clusters are not magnetically active, their structure does show an interesting dependence with their magnetic moment. This is worth noting because research has predicted that gold cluster have an ability to adsorb small molecules (including important biomolecules) that is very dependent on slight structural changes. We have found a possible way that this could happen: the small structural changes induced by a small shift in the magnetic moment produce relatively large charge transfers from certain atoms to others, which changes the reaction loci. On the other hand, it is reasonable to expect that an external magnetic field could induce said changes, as it is a likely source of the change of the magnetic state of the cluster. This gives us a degree of control over the potential capability of adsorption of the clusters, and also makes the 8 atom gold clusters into candidates to show magnetostriction phenomena, of great utility in several fields.
Interaction between gold clusters of different sizes with the oxygen molecule.
We know that Au clusters can have a large magnetic moment and are likely to show interesting magnetostriction phenomena. This, together with their biochemical catalytic capabilities, makes them good candidates to operate as vectors in medical applications. But if gold clusters are to be used in a biological medium, it is necessary to know if they are as inert as they are in bulk state, and which conditions might cause them to vary their geometry. A cluster with an adequate catalytic ability and the right magnetostriction, under reasonably low magnetic field values, could be completely useless when the effect of oxidization is taken into account.
Oxidation of small Au clusters has been studied in the past, but it has always focused on oxygen molecules placed in the same plane as the bidimensional clusters. We are going to attempt to find if other off-plane positions are possible for these bidimensional Au clusters, and establish if these structures are stable or not. Other points of interest will be: i) finding if the structure of the pure Au cluster is distorted due to the interaction with the O2 molecule; ii) whether the HOMO-LUMO gap is a valid method to determine the stability of these clusters. The distortion is relevant because the catalytic properties of Au clusters are highly sensitive to small torsions of their planar surface.
Our calculations led us to find that the interaction between oxygen molecules and small, bidimensional Au clusters depends on the Au cluster having even or odd number of atoms. For the even ones, the O2 bonds in the cluster plane, while for the odd ones, it bonds off-plane. This off-plane interaction does not create an actual chemical bond, but a sum of magnetostatic and electrostatic weak bonds, which provide great stability. Said bonding also distorts the original structure and gives it a slight curvature. For our purposes, this means that ab initio calculations are indeed an effective tool to predict whether a cluster, or any kind of molecule, can be suitable to work in a biotic environment. Au small clusters with an even number of atoms will have their electronic properties changed by the bond with the O2 molecule, while the ones with an odd amount will remain in the same electronic configuration. Nevertheless the possible catalytic properties will be affected because of their torsion in presence of oxygen.
Disagreement between theory and experiment: bismuth-manganese clusters.
Atoms in bare small clusters can exhibit far larger MMs than when isolated or within crystals. This effect can be enhanced when impurities are added. A question worth answering was whether this effect would be enhanced or muffled when the amount of impurities is as high as to constitute an alloy, and not simply a cluster with impurities. The study of transition metal binary clusters was triggered by the exceptionally high magnetism found in CoRh nanoparticles by Zitoun and collaborators. Since then, a number of magnetically enhanced nanoalloys of ferromagnetic and non-magnetic transition metals have been studied theoretically and experimentally. Yin and collaborators found an enhanced magnetism in BiMn clusters for Bi-to-Mn ratios close to 2 in their Stern-Gerlach measurements. In a latter paper, Chen and collaborators performed extensive density functional theory (DFT) calculations on BinMnm (n=1-6, m=1-12) clusters to learn their structure and how their magnetism works. While they have found quite a good agreement with the experiment for this series, there are a few large discrepancies between the theoretical value of the magnetic moment and the measured one. The most significant one is the Bi4Mn, where DFT calculations predict a total MM ¿T = 5 ¿B while the experiment measures 1.6 ¿B. The cause of this discrepancy was not found, although extensive geometric optimizations and an estimation of the orbital contribution to the magnetic moment were performed to solve this problem.
We found that three isomers among the 21 most stable actually have a total magnetic moment below the experimental one, but these isomers are too high in energy respect to the lowest-lying one. Furthermore, there were 18 isomers with lower energy, so these three could not make up for a fraction of the population of randomly created Bi4Mn clusters significant enough to reduce the average total magnetic moment down to the experimental value. We had found that any of the used XC functional approximations does not make the ground state of the lowest-lying isomer have a lower magnetic moment. The analysis of the positive and negative singly-charged ions had also yielded no lowest-energy structure with a magnetic moment close to the experimental one. In fact, none of the ions of the 21 isomers we have considered has a ground state with a magnetic moment lower than the experiment. This should be expected though, as the only lower magnetic state available to ions of Bi4Mn is a singlet. The possible presence of ions would not help us explain the experiment unless we had found ground states at ¿T = 0 ¿B. From our non-collinear calculation, we rule out other two possible sources of the disagreement: i) the magnetic configuration is collinear to a high degree, even when allowing the individual atomic magnetic moment to arrange freely, and still produces states with high magnetic moment: between 4 and 6.4 ¿B, hence this cannot be the source. ii) The orbital magnetic moment is too low in absolute value compared to the total magnetic moment to produce a significant reduction in it.
All these consistently negative results in our search for a source of the discrepancy in the calculations suggest that said source is not an actual error of the calculations.
Furthermore, the possibility that DFT method itself is not reliable to study this cluster is unlikely: the DFT results are in fairly good agreement for almost all the other BinMnm clusters. Nevertheless arriving at a negative result is not good enough. We want to know if we can rely on ab initio calculations as a tool, not just to describe what is already experimentally known, but to know why the experimental results are the ones they are.
Did the obtained theoretical data give us any clue as to why this disagreement happens that is not explicit? We suggested that the experimental sample of Bi4Mn could be composed of a population of different structures. This would be the most obvious explanation as to why the magnetic moment of the experimental clusters is not integer. Would it possible that some of our structures 19, 20, 21 made up a significant fraction of said sample? We had found no reason why DFT would increase the energy of these structures specifically, so the only other option, for that experimental sample composition to be true, is that a factor in the experiment favoured their production. We thought it was remarkable that the structures 19, 20, 21 were very similar the lowest-lying isomer for Bi3Mn clusters (a tetrahedron), and could be obtained from them by simply adding an extra Bi atom to the Mn end of the tetrahedron. If Bi3Mn clusters formed much more quickly in the experimental device than Bi4Mn ones, it would not be unreasonable to think that the later would form from the former. Bi3Mn also has a large electrostatic dipole we have calculated to be 2.4 D in the direction that connects the Mn atom with the centre of the triangle formed by the three Bi atoms. This dipolar moment makes them highly stable, and could help a fourth Bi atom to couple to the Mn, instead of breaking up the tetrahedron to form the calculated Bi4Mn lowest-lying structure. If the process of measurement of mass and magnetic moment of the clusters is fast enough, or if the energy barrier between these mono-capped tetrahedron structure and the lowest laying isomer is high enough, they might not have enough time to relax into the said lowest-lying isomer during their flight in the Stern-Gerlach device (the flight time is estimated in around 10 ms). This would result in some of our structures 19, 20 or 21 making up a large fraction of the measured Bi4Mn clusters.
To verify this, the path we think it would be most viable is perform an analysis of the optical properties of the experimentally obtained Bi4Mn clusters to identify their geometry, and see if they match with the isomers theoretically obtained as the lowest energy ones, or instead they match with any of the 19, 20, 21 isomers that actually show a low total magnetic moment. If the later case were true, then we would have to explain why the experimental setup produces structures that theoretically at T= 0 K are known not to be the fundamental one.
This explanation is not mere speculation: it shows that the task of the ab initio calculations is not just explaining the experimental results, or predicting results where experiments cannot (yet) be made. When problems arise, ab initio calculations also should be able to lead the experiment in the right direction, to find the solution.
Characterization of gold nanoparticles samples.
Clusters made up by a much larger number of atoms are called nanoparticles (NPs), and they do not show just cluster-like properties. When the amount of atoms of a metallic cluster is counted by thousands instead of dozens, the properties that the metal shows in bulk state begin to manifest, mixed with true cluster-like properties: this region of sizes is called the mesoscale.
This mixture of properties, together with the possibility to tailor their shape and size have made the noble-metal NPs specially attractive to the applied fields of catalysis, photonics, electronics, sensing, biolabeling and imaging. For example, i) the number, location, and intensity of surface plasmon bands in Au and Ag nanocrystals exhibits a strong correlation with the particle shape which makes these nanoparticles (NPs) useful as therapeutic agents due to surface-plasmon absorption in the near-infrared spectral region, or as biosensors when absorption takes place in the visible wavelength range; ii) the presence of sharp edges and tips has been shown to generate an electric-field enhancement, which is importantfor applications involving metal nanoparticles as sensors; iii) the reactivity and selectivity of metal nanocatalysts depend strongly on the crystallographic planes exposed on the particles surface and, thus, may be tuned by controlling the particle morphology; iv) additionally, nanoparticle morphology will ultimately determine the way in which nanoparticles can be assembled. Hence, the ability to control the size and shape of noble metal NPs is highly desiderable.
NPs are large enough that there is a relative ease to produce them, and we can rely on many strategies: photochemistry, thermochemistry, wet-chemistry, and biochemistry.
Materials that are so readily available through experimental means are ideal to compare with theoretical calculations. The main drawback is that the study of these systems cannot be realistically performed, nowadays, through ab-initio techniques, not without extreme approximations. In this case we are forced to fall back to a different, but much more commonly used, computational method: computer-assisted modelization. An experimental physics group of USC colleagues has recently found a method to reliably control the shape and size of Au NPs by choosing an appropriate solvent in which the NPs grow.
While characterization of the shape of these NP can be done by means of electron microscopy, a reliable study of the experimental sample size distribution should include the analysis of its optical spectrum. Through our model calculations we can obtain theoretical spectra for each size and geometry of the nanoparticles, and then these spectra are weighed with the measured size distribution to observe if the spectrum fits with the experimental one, and ensure that the measured size distribution is correct.
We were able to obtain a very accurate calculation of what the extinction spectra of large samples of both bipyramids and triangular nanoplates should look like. We achieved this by using a very simple modelization of how gold nanoparticles work: simply assuming they can be substituted by a cubic lattice of point dipoles. We have also used rough estimations of the values of dielectric function of the aqueous media the nanoparticles were submerged in. Despite all of this, the results happen to be very accurate compared to the experimentally obtained spectra, which means that this method would have a very high possibility of success if it were relied on as the sole method to characterize experimental nanoparticle samples.
