Axial symmetry and transverse trace-free tensors in numerical relativity

Transverse trace-free (TT) tensors play an important role in the initial conditions of numerical relativity, containing two of the component freedoms. Expressing a TT tensor entirely, by the choice of two scalar potentials, is not a trivial task however. Assuming the added condition of axial symmetry, expressions are given in both spherical and cylindrical coordinates, for TT tensors in flat space. A coordinate relation is then calculated between the scalar potentials of each coordinate system. This is extended to a non-flat space, though only one potential is found. The remaining equations are reduced to form a second order partial differential equation in two of the tensor components. With the axially symmetric flat space tensors, the choice of potentials giving Bowen-York conformal curvatures, are derived. A restriction is found for the potentials which ensure an axially symmetric TT tensor, which is regular at the origin, and conditions on the potentials, which give an axially symmetric TT tensor with a spherically symmetric scalar product, are also derived. A comparison is made of the extrinsic curvatures of the exact Kerr solution and numerical Bowen-York solution for axially symmetric black hole space-times. The Brill wave, believed to act as the difference between the Kerr and Bowen-York space-times, is also studied, with an approximate numerical solution found for a mass-factor, under different amplitudes of the metric.

Ab initio calculations of group 4 metallocene reaction mechanisms: atomic layer deposition and bond activation catalysis

Thin film dielectrics based on titanium, zirconium or hafnium oxides are being introduced to increase the permittivity of insulating layers in transistors for micro/nanoelectronics and memory devices. Atomic layer deposition (ALD) is the process of choice for fabricating these films, as it allows for high control of composition and thickness in thin, conformal films which can be deposited on substrates with high aspect-ratio features. The success of this method depends crucially on the chemical properties of the precursor molecules. A successful ALD precursor should be volatile, stable in the gas-phase, but reactive on the substrate and growing surface, leading to inert by-products. In recent years, many different ALD precursors for metal oxides have been developed, but many of them suffer from low thermal stability. Much promise is shown by group 4 metal precursors that contain cyclopentadienyl (Cp = C5H5-xRx) ligands. One of the main advantages of Cp precursors is their thermal stability. In this work ab initio calculations were carried out at the level of density functional theory (DFT) on a range of heteroleptic metallocenes [M(Cp)4-n(L)n], M = Hf/Zr/Ti, L = Me and OMe, in order to find mechanistic reasons for their observed behaviour during ALD. Based on optimized monomer structures, reactivity is analyzed with respect to ligand elimination. The order in which different ligands are eliminated during ALD follows their energetics which was in agreement with experimental measurements. Titanocene-derived precursors, TiCp*(OMe)3, do not yield TiO2 films in atomic layer deposition (ALD) with water, while Ti(OMe)4 does. DFT was used to model the ALD reaction sequence and find the reason for the difference in growth behaviour. Both precursors adsorb initially via hydrogen-bonding. The simulations reveal that the Cp* ligand of TiCp*(OMe)3 lowers the Lewis acidity of the Ti centre and prevents its coordination to surface O (densification) during both of the ALD pulses. Blocking this step hindered further ALD reactions and for that reason no ALD growth is observed from TiCp*(OMe)3 and water. The thermal stability in the gas phase of Ti, Zr and Hf precursors that contain cyclopentadienyl ligands was also considered. The reaction that was found using DFT is an intramolecular α-H transfer that produces an alkylidene complex. The analysis shows that thermal stabilities of complexes of the type MCp2(CH3)2 increase down group 4 (M = Ti, Zr and Hf) due to an increase in the HOMO-LUMO band gap of the reactants, which itself increases with the electrophilicity of the metal. The reverse reaction of α-hydrogen abstraction in ZrCp2Me2 is 1,2-addition reaction of a C-H bond to a Zr=C bond. The same mechanism is investigated to determine if it operates for 1,2 addition of the tBu C-H across Hf=N in a corresponding Hf dimer complex. The aim of this work is to understand orbital interactions, how bonds break and how new bonds form, and in what state hydrogen is transferred during the reaction. Calculations reveal two synchronous and concerted electron transfers within a four-membered cyclic transition state in the plane between the cyclopentadienyl rings, one π(M=X)-to-σ(M-C) involving metal d orbitals and the other σ(C-H)-to-σ(X-H) mediating the transfer of neutral H, where X = C or N. The reaction of the hafnium dimer complex with CO that was studied for the purpose of understanding C-H bond activation has another interesting application, namely the cleavage of an N-N bond and resulting N-C bond formation. Analysis of the orbital plots reveals repulsion between the occupied orbitals on CO and the N-N unit where CO approaches along the N-N axis. The repulsions along the N-N axis are minimized by instead forming an asymmetrical intermediate in which CO first coordinates to one Hf and then to N. This breaks the symmetry of the N-N unit and the resultant mixing of MOs allows σ(NN) to be polarized, localizing electrons on the more distant N. This allowed σ(CO) and π(CO) donation to N and back-donation of π*(Hf2N2) to CO. Improved understanding of the chemistry of metal complexes can be gained from atomic-scale modelling and this provides valuable information for the design of new ALD precursors. The information gained from the model decomposition pathway can be additionally used to understand the chemistry of molecules in the ALD process as well as in catalytic systems.

Multi-scale modelling of atomic layer deposition

High-permittivity ("high-k") dielectric materials are used in the transistor gate stack in integrated circuits. As the thickness of silicon oxide dielectric reduces below 2 nm with continued downscaling, the leakage current because of tunnelling increases, leading to high power consumption and reduced device reliability. Hence, research concentrates on finding materials with high dielectric constant that can be easily integrated into a manufacturing process and show the desired properties as a thin film. Atomic layer deposition (ALD) is used practically to deposit high-k materials like HfO2, ZrO2, and Al2O3 as gate oxides. ALD is a technique for producing conformal layers of material with nanometer-scale thickness, used commercially in non-planar electronics and increasingly in other areas of science and technology. ALD is a type of chemical vapor deposition that depends on self-limiting surface chemistry. In ALD, gaseous precursors are allowed individually into the reactor chamber in alternating pulses. Between each pulse, inert gas is admitted to prevent gas phase reactions. This thesis provides a profound understanding of the ALD of oxides such as HfO2, showing how the chemistry affects the properties of the deposited film. Using multi-scale modelling of ALD, the kinetics of reactions at the growing surface is connected to experimental data. In this thesis, we use density functional theory (DFT) method to simulate more realistic models for the growth of HfO2 from Hf(N(CH3)2)4/H2O and HfCl4/H2O and for Al2O3 from Al(CH3)3/H2O.Three major breakthroughs are discovered. First, a new reaction pathway, ’multiple proton diffusion’, is proposed for the growth of HfO2 from Hf(N(CH3)2)4/H2O.1 As a second major breakthrough, a ’cooperative’ action between adsorbed precursors is shown to play an important role in ALD. By this we mean that previously-inert fragments can become reactive once sufficient molecules adsorb in their neighbourhood during either precursor pulse. As a third breakthrough, the ALD of HfO2 from Hf(N(CH3)2)4 and H2O is implemented for the first time into 3D on-lattice kinetic Monte-Carlo (KMC).2 In this integrated approach (DFT+KMC), retaining the accuracy of the atomistic model in the higher-scale model leads to remarkable breakthroughs in our understanding. The resulting atomistic model allows direct comparison with experimental techniques such as X-ray photoelectron spectroscopy and quartz crystal microbalance.

Atomic layer deposition of copper for CMOS interconnects

This work concerns the atomic layer deposition (ALD) of copper. ALD is a technique that allows conformal coating of difficult topographies such as narrow trenches and holes or even shadowed regions. However, the deposition of pure metals has so far been less successful than the deposition of oxides except for a few exceptions. Challenges include difficulties associated with the reduction of the metal centre of the precursor at reasonable temperatures and the tendency of metals to agglomerate during the growth process. Cu is a metal of special technical interest as it is widely used for interconnects on CMOS devices. These interconnects are usually fabricated by electroplating, which requires the deposition of thin Cu seed layers onto the trenches and vias. Here, ALD is regarded as potential candidate for replacing the current PVD technique, which is expected to reach its limitations as the critical dimensions continue to shrink. This work is separated into two parts. In the first part, a laboratory-scale ALD reactor was constructed and used for the thermal ALD of Cu. In the second part, the potentials of the application of Cu ALD on industry scale fabrication were examined in a joint project with Applied Materials and Intel. Within this project precursors developed by industrial partners were evaluated on a 300 mm Applied Materials metal-ALD chamber modified with a direct RF-plasma source. A feature that makes ALD a popular technique among researchers is the possibility to produce high- level thin film coatings for micro-electronics and nano-technology with relatively simple laboratory- scale reactors. The advanced materials and surfaces group (AMSG) at Tyndall National Institute operates a range of home-built ALD reactors. In order to carry out Cu ALD experiments, modifications to the normal reactor design had to be made. For example a carrier gas mechanism was necessary to facilitate the transport of the low-volatile Cu precursors. Precursors evaluated included the readily available Cu(II)-diketonates Cu-bis(acetylacetonate), Cu-bis(2,2,6,6-tetramethyl-hepta-3,5-dionate) and Cu-bis(1,1,1,5,5,5-hexafluoacetylacetonate) as well as the Cu-ketoiminate Cu-bis(4N-ethylamino- pent-3-en-2-onate), which is also known under the trade name AbaCus (Air Liquide), and the Cu(I)- silylamide 1,3-diisopropyl-imidazolin-2-ylidene Cu(I) hexamethyldisilazide ([NHC]Cu(hmds)), which was developed at Carleton University Ottawa. Forming gas (10 % H2 in Ar) was used as reducing agent except in early experiments where formalin was used. With all precursors an extreme surface selectivity of the deposition process was observed and significant growth was only achieved on platinum-group metals. Improvements in the Cu deposition process were obtained with [NHC]Cu(hmds) compared with the Cu(II) complexes. A possible reason is the reduced oxidation state of the metal centre. Continuous Cu films were obtained on Pd and indications for saturated growth with a rate of about 0.4 Å/cycle were found for deposition at 220 °C. Deposits obtained on Ru consisted of separated islands. Although no continuous films could be obtained in this work the relatively high density of Cu islands obtained was a clear improvement as compared to the deposits grown with Cu(II) complexes. When ultra-thin Pd films were used as substrates, island growth was also observed. A likely reason for this extreme difference to the Cu films obtained on thicker Pd films is the lack of stress compensation within the thin films. The most likely source of stress compensation in the thicker Pd films is the formation of a graded interlayer between Pd and Cu by inter-diffusion. To obtain continuous Cu films on more materials, reduction of the growth temperature was required. This was achieved in the plasma assisted ALD experiments discussed in the second part of this work. The precursors evaluated included the AbaCus compound and CTA-1, an aliphatic Cu-bis(aminoalkoxide), which was supplied by Adeka Corp.. Depositions could be carried out at very low temperatures (60 °C Abacus, 30 °C CTA-1). Metallic Cu could be obtained on all substrate materials investigated, but the shape of the deposits varied significantly between the substrate materials. On most materials (Si, TaN, Al2O3, CDO) Cu grew in isolated nearly spherical islands even at temperatures as low as 30 °C. It was observed that the reason for the island formation is the coalescence of the initial islands to larger, spherical islands instead of forming a continuous film. On the other hand, the formation of nearly two-dimensional islands was observed on Ru. These islands grew together forming a conductive film after a reasonably small number of cycles. The resulting Cu films were of excellent crystal quality and had good electrical properties; e.g. a resistivity of 2.39 µΩ cm was measured for a 47 nm thick film. Moreover, conformal coating of narrow trenches (1 µm deep 100/1 aspect ratio) was demonstrated showing the feasibility of the ALD process.

Cooperation between adsorbates accounts for the activation of atomic layer deposition reactions

Atomic layer deposition (ALD) is a technique for producing conformal layers of nanometre-scale thickness, used commercially in non-planar electronics and increasingly in other high-tech industries. ALD depends on self-limiting surface chemistry but the mechanistic reasons for this are not understood in detail. Here we demonstrate, by first-principle calculations of growth of HfO2 from Hf(N(CH3)2)4–H2O and HfCl4–H2O and growth of Al2O3 from Al(CH3)3–H2O, that, for all these precursors, co-adsorption plays an important role in ALD. By this we mean that previously-inert adsorbed fragments can become reactive once sufficient numbers of molecules adsorb in their neighbourhood during either precursor pulse. Through the calculated activation energies, this ‘cooperative’ mechanism is shown to have a profound influence on proton transfer and ligand desorption, which are crucial steps in the ALD cycle. Depletion of reactive species and increasing coordination cause these reactions to self-limit during one precursor pulse, but to be re-activated via the cooperative effect in the next pulse. This explains the self-limiting nature of ALD.

Langmuir-Blodgett assembly of colloidal crystals combined with controlled infiltration of conducting metal oxides

Colloidal photonic crystals (PhCs) possess a periodic dielectric structure which gives rise to a photonic band gap (PBG) and offer great potential in the ability to modify or control light at visible wavelengths. Although the refractive index contrast between the void or infill and the matrix material is paramount for photonics applications, integration into real optoelectronics devices will require a range of added functionalities such as conductivity. As such, colloidal PhCs can be used as templates to direct infiltration of other functional materials using a range of deposition strategies. The work in this thesis seeks to address two challenges; first to develop a reproducible strategy based on Langmuir-Blodgett (LB) deposition to assemble high quality colloidal PhCs based on silica with precise film thickness as most other assembly methods suffer from a lack of reproducibility thickness control. The second is to investigate the use of LBdeposited colloidal PhCs as templates for infiltration with conducting metal oxide materials using vapor phase deposition techniques. Part of this work describes the synthesis and assembly of colloidal silica spheres with different surface chemical functionalities at the air-water interface in preparation for LB deposition. Modification of surface funtionality conferred varying levels of hydrophobicity upon the particles. The behaviour of silica monolayer films at the air-water interface was characterised by Brewster Angle Microscopy and surface pressure isotherms with a view to optimising the parameters for LB deposition of multilayer colloidal PhC films. Optical characterisation of LB-fabricated colloidal PhCs indicated high quality photonic behaviour, exhibiting a pseudo PBG with a sharp Bragg diffraction peak in the visible region and reflectance intensities greater than 60%. Finally the atomic layer deposition (ALD) of nominally undoped ZnO and aluminium “doped” ZnO (Al-doped ZnO) inside the pores of a colloidal PhC assembled by the LB technique was carried out. ALD growth in this study was performed using trimethyl aluminium (TMA) and water as precursors for the alumina and diethyl zinc (DEZn) and water for the ZnO. The ZnO:Al films were grown in a laminate mode, where DEZn pulses were substituted for TMA pulses in the sequences with a Zn:Al ratio 19:1. The ALD growth of ZnO and ZnO:Al in colloidal PhCs was shown to be highly conformal, tuneable and reproducible whilst maintaining excellent photonic character. Furthermore, at high levels of infiltration the opal composite films demonstrated significant conductivity.