Corrected Effective Medium Research Articles

Improved model-based infrared reflectrometry for measuring deep trench structures

Model-based infrared reflectrometry (MBIR) has been introduced recently for characterization of high-aspect-ratio deep trench structures in microelectronics. The success of this technique relies heavily on accurate modeling of trench structures and fast extraction of trench parameters. In this paper, we propose a modeling method named corrected effective medium approximation (CEMA) for accurate and fast reflectivity calculation of deep trench structures. We also develop a method combining an artificial neural network (ANN) and a Levenberg-Marquardt (LM) algorithm for robust and fast extraction of geometric parameters from the measured reflectance spectrum. The simulation and experimental work conducted on typical deep trench structures has verified the proposed methods and demonstrated that the improved MBIR metrology achieves highly accurate measurement results as well as fast computation speed.

Structures and energetics of Ni24–Ni55 clusters

We predict stable geometrical structures and interaction energies of Ni clusters using non self-consistent electron density functional based corrected effective medium (CEM) and MD/MC-CEM methods. A plot of the reaction energies for the atomic ejection process, NiN+1→NiN+Ni, for Ni24–Ni55 displays a number of informative characteristics: (a) peaks and valleys represent internal structural rearrangement in which the number of core atoms increases by at least one and; (b) a plateau at N=50–54 is associated with the closing of the second MacKay icosahedron at Ni55. The lowest energy structures of NiN clusters for N=24–55 are dissimilar generally to those of both rare gas clusters and fragments of the bulk crystal lattice except where a stable icosahedral or bulk core is present. The growth scheme for N<50 is determined by the stability and structure of the changing number of core atoms. By contrast, the growth scheme for 51≤N≤55 is determined by the addition of surface atoms to a very stable and invariant 13-atom icosahedral core. The theoretical predictions are compared to available model growth schemes and experimental data.

Self-diffusion of adatoms on fcc(110) surfaces

We investigate the mechanisms and associated energies of diffusion for the Ni Ni(110) , Cu Cu(110) , Pd Pd(110) and Ag Ag(110) systems using the corrected effective medium (CEM) method. On the (110) surface, cross-channel diffusion can occur via atomic exchange of the adsorbate atom with an atom of the channel wall or by hopping over the short-bridge site located in the channel wall. We find that the barrier for cross-channel diffusion is much smaller for the atomic exchange path. In-channel diffusion occurs via hopping over the long-bridge site. We find that the activation barrier associated with in-channel diffusion is smaller than the activation barrier associated with cross-channel diffusion by 0.17, 0.01, 0.03 and 0.08 eV for Ni, Cu, Pd and Ag, respectively. We also investigate convergence of the activation barriers by varying the number of active atoms in the simulations. Decreases of 0.3–0.6 eV are found as the number of active atoms is increased from nearest neighbors to a (converged) large number of active atoms; this is due to relaxation of the strain. The decreases are nearly identical for the in-channel and cross-channel mechanisms.

Molecular dynamics and kinetic Monte Carlo simulations of Fe island growth on Cu(111)

We have performed extensive molecular dynamics (MD) simulations of a 0.07 ML coverage of Fe adatoms on Cu(111) at 300 K using a many-body interaction potential from the molecular dynamics/Monte-Carlo corrected effective medium theory. The Fe atoms were deposited randomly and then followed for two hundred picoseconds (ps) while annealing at a temperature of 300 K. The results of these MD simulations were analyzed to understand the general features of island growth. One finding was that islands as large as four atoms are only one of magnitude less mobile than monomers. Thus, the mobility of small islands played a significant role in the growth of large islands. In addition, the presence of next-nearest neighbor attractions was found to increase the speed of this growth. Another finding was that the Fe islands obtained their equilibrium structures only very slowly compared to the times of aggregation or coalescence. The MD results were used to construct and test several kinetic Monte Carlo models that incorporated the newly identified processes. Considering these MC models, we discuss some details of island structure, stability and aggregation.

Theoretical predictions of heats of formation of fcc binary transition-metal alloys using the corrected effective-medium theory

We present heats of formation of ordered fcc transition-metal binary alloys as predicted by the non-self-consistent electron density functional corrected effective-medium (CEM) theory. Two forms of the CEM theory are used. The first includes explicit numerical evaluation of the kinetic-exchange-correlation energy functional, a computationally intensive task. The second approximates this term as a local function of electron density and leads to a formulation that is very similar to the embedded-atom method. Calculated values of the energies of formation from the two forms of the CEM method are compared to each other and to experimental data. We demonstrate that both forms of the CEM theory provide formation energies accurate to within about 0.05--0.1 eV/atom (5--10 kJ/mol) for 3[ital d] and 4[ital d] binary alloys.

Surface segregation in bimetallic clusters: Predictions using a molecular dynamics/Monte Carlo corrected effective medium theory

Computer simulations were performed to study the surface composition of Rh x Pd 1 − x , Rh x Ni 1 − x and Ni x Pd 1 − x clusters where 0.1 ⩽ x ⩽ 0.9. Clusters containing 201 and 1289 atoms were simulated at 200, 600 and 1000 K. The simulation consists of the Monte Carlo (MC) exchange of the positions of unlike atoms and the molecular-dynamics (MD) evolution of all atoms between exchanges. A corrected effective medium (CEM) theory based interatomic potential, with acronym MD/MC-CEM, was used to generate the forces. Our simulations indicate that the metal with lower surface energy segregates first to the catalytically very important edge-corner sites, then to other surface sites. Increasing the temperature decreases the degree of segregation at edge-corner sites more strongly than on the entire surface. Interestingly, NiPd exhibits stronger surface segregation but weaker edge-corner segregation than RhNi. Results for other 201- and 1289-atom 50%-50% bimetallic clusters at 600 K, selected from Rh, Ni, Pd, Cu and Ag, are also presented. In particular, the large lattice-size mismatch in NiAg and CuAg invalidates the simple geometrical constraints on surface segregation that hold for other bimetallic clusters.

Driving force for surface segregation in bimetallic catalysts

The catalytic properties of bimetallic clusters are influenced by the detailed structure of the cluster surface. We briefly describe two methods developed to treat this problem, from the computationally demanding corrected effective medium (CEM) theory (a non-self-consistent density functional based method) to the computationally simple method of surface modified pair potentials (SMPP). Parametrization of the latter is accomplished using the former. Comparisons of theoretical predictions with experimental data are made for the heat of formation in Rh x Pt 1− x alloys and for surface segregation behavior in Rh 0.9Pt 0.1(111). Results on the shape, site composition and surface micromixing are shown for 201 atom clusters (dispersion of 0.6) and Rh x Pt 1− x (111). A fundamental point follows from the fact that the CEM calculated cohesive energy of Rh is slightly smaller than that of Pt, but the surface energy of Rh is significantly larger: the driving force for surface segregation is the relative surface energies not the relative cohesive energies. In general, one can predict surface energy differences by properly accounting for the variation of the bond energy with coordination.

Driving force for surface segregation in bimetallic catalysts

The catalytic properties of bimetallic clusters are influenced by the detailed structure of the cluster surface. We briefly describe two methods developed to treat this problem, from the computationally demanding corrected effective medium (CEM) theory (a non-self-consistent density functional based method) to the computationally simple method of surface modified pair potentials (SMPP). Parametrization of the latter is accomplished using the former. Comparisons of theoretical predictions with experimental data are made for the heat of formation in Rh x Pt 1− x alloys and for surface segregation behavior in Rh 0.9Pt 0.1(111). Results o shape, site composition and surface micromixing are shown for 201 atom clusters (dispersion of 0.6) and Rh x Pt 1− x (111). A fundamental point follows from the fact that the CEM calculated cohesive energy of Rh is slightly smaller than that of Pt, but the surface energy of Rh is significantly larger: the driving force for surface segregation is the relative surface energies not the relative cohesive energies. In general, one can predict surface energy differences by properly accounting for the variation of the bond energy with coordination.

The structure of NiN and PdN clusters: 4≤N≤23

Stable geometrical structures of NiN and PdN clusters (N=4–23) are identified using a corrected effective medium (CEM) theory. Structural optimization is accomplished by simulated annealing using analytic derivatives to determine the interatomic forces. Unique structural features of these metal clusters are noted, especially in relation to the bulk and surface phases of these metals and to structures commonly associated with rare gas clusters. Elucidation of the general features of cluster growth leads to the principle that transition metal clusters generally maximize the minimum coordination of any atom. By contrast, rare gas clusters maximize the number of interatomic distances close to the optimal distance for the pairwise interaction between rare gas atoms. The latter can be interpreted as the packing of hard balls. Structural transformations between isomers of similar energy are also examined for selected sizes.

Metal/metal homoepitaxy on fcc(111) and fcc(001) surfaces: Deposition and scattering from small islands

For systems of a single chemical species, we investigate the dynamics of adsorption of a metal atom on a clean fcc(111) metal substrate and on a pyramidal defect built on the fcc(001) substrate; the defect extends three layers above the surface, with nine, four, and one atoms in each layer, respectively. Each face thus forms a small part of an fcc(111) face. Molecular dynamics (MD) simulations are used where the interaction energy and the corresponding forces are generated from the recently developed non-self-consistent density functional based corrected effective medium (CEM) method in its simplest and computationally most efficient semiempirical MD/MC-CEM formulation. In previous work, the MD/MC-CEM form has been shown to predict the geometric and energetic properties of clean and adsorbate covered surfaces accurately. We show that there is no transient mobility for the Cu/Cu(111) system at a surface temperature of 80 K and an incident gas atom kinetic energy of 0.25 eV at incident angles up to 45° from the surface normal. This finding should hold for other homogeneous metal deposition systems since it arises from the fact that energy dissipation to the metal substrate is so efficient. For the Cu system, we determine that the cross section for sticking on the sides of the 14 atom pyramid varies from 9.0 to 37.2 Å2 for two MD/MC-CEM based potential energy surfaces (PES) differing only in the energetics of low coordination binding. Normalized to the aiming area, these give probabilities of 8.6% to 35.7%. Thus, it is shown that the low coordination region of the interaction strongly impacts the determination of growth modes, shifting from a funneling method giving two-dimensional (2D) layer by layer growth to a 3D mode, producing a rougher surface.

A non-unique relationship between potential energy surface barrier and dynamical diffusion barrier: fcc (111) metal surface

We present results from molecular dynamics simulations of the diffusion of a single metal atom on the surface of a fcc (111) metal. The central point that is addressed is the relationship between features of the potential energy surface and the diffusion rate when the surface (energy) corrugation is small. The systems considered here are Ag and Ag(111) and Rh on Rh(111). The interatomic forces were predicted using an approximation to the many-body non-self-consistent density functional based corrected effective medium theory (with acronym MD/MC-CEM), and an empirical Morse potential energy surface. Both potential energy forms are completely specified by the properties of the bulk systems, with the former also predicting very good values of surface energies on low Miller index surfaces and diffusion barriers on fcc (001) surfaces. Diffusion rates were extracted from the long time behavior of the adatom's mean square displacement and were found to obey an Arrhenius-type dependence on temperature. Good agreement with experimental diffusion rates was found with both potential energy surfaces. A different potential energy surface with much larger energy corrugation also yields agreement with experiment. We conclude that diffusion on these atomically smooth surfaces is governed by correlated dynamical behavior which eliminates the possibility of a unique extraction of the corrugation profile from experimental data.

Corrected effective-medium study of metal-surface relaxation.

We have studied the relaxation of some metal surfaces using a corrected effective-medium (CEM) theory. CEM is a non-self-consistent, density-functional technique. The interaction energy is calculated as a sum of three components: the embedding energy of an atom in jellium, the interatomic Coulomb energy, and the kinetic exchange-correlation energy. A theoretical procedure has been developed to determine the embedding energies via linear-muffin-tin-orbital calculations. This refinement of the theory is tested for surface energies and structures of some relaxed Al, Ni, Cu, Rh, Pd, and Ag surfaces. Comparisons are made to the embedded-atom method, the effective-medium technique, and a simpler version of CEM that is used in molecular-dynamics and Monte Carlo simulations. We present an in-depth analysis of the results, and discuss the physical basis of surface relaxation within CEM. Finally, we address the limitations inherent in calculations of metal-surface relaxation.

Potential energy hypersurfaces for the interaction of NO with the Ag(111) surface

We present a new potential energy surface for the interaction of the NO molecule with a close-packed Ag(111) surface obtained using the recently developed corrected effective medium (CEM) method. The dual NO–Ag(111) interaction potentials, which reflect the electronic orbital degeneracy of the NO molecule in its ground(X 2Π) electronic state, were determined at a large variety of distances and orientations of the NO molecule above the surface, and for approach of the molecule over three different sites on the surface unit cell, described by a cluster of greater than 10 Ag atoms. The more than 600 calculated interaction energies were then fit to a compact analytic form, which allows the determination of the interaction potential at any position above the surface. The strongest binding corresponds to a nearly flat orientation of the NO molecule, with approach of the N end slightly preferred, and with the single antibonding π orbital of NO pointing toward the surface. In this geometry, the binding energy above, respectively, the atop, two-fold bridge, and three-fold center sites site is 2200, 1773, and 1574 cm−1. By contrast when the antibonding π electron is oriented in a plane parallel to the surface, the binding energy diminishes dramatically, and the dependence of the interaction energy on the orientation of the NO molecule is also diminished. Here, the three-fold center site is preferred, but with a well depth of only 885 cm−1 (0.110 eV).

Theoretical studies of dynamical phenomena in epitaxial surface systems

Abstract The increasing resolution of scanning tunneling microscopy techniques is beginning to reveal new interesting aspects of atomic behavior at surfaces. The ability to monitor diffusion of even single atoms is one dynamical example. This atomic scale resolution enables a close connection to theory, which plays an important role in uncovering the driving forces and mechanisms behind these dynamical events. In this paper we briefly discuss the theoretical foundations of the corrected effective medium (CEM) method and its application to the epitaxy of metals on metals. Static interaction energy and dynamic Monte Carlo (MC) and molecular dynamics (MD) calculations are employed to examine equilibrium structures and the mechanisms in which equilibrium is obtained. Three specific applications are considered: (1) the mechanism for the formation of the ordered c(2 × 2) Au/Cu(100) surface alloy; (2) The formation of Ag/Rh/Ag(100) “sandwich” structures; (3) The growth of Au on Ag(110).

Theoretical studies of dynamical phenomena in epitaxial surface systems

The increasing resolution of scanning tunneling microscopy techniques is beginning to reveal new interesting aspects of atomic behavior at surfaces. The ability to monitor diffusion of even single atoms is one dynamical example. This atomic scale resolution enables a close connection to theory, which plays an important role in uncovering the driving forces and mechanisms behind these dynamical events. In this paper we briefly discuss the theoretical foundations of the corrected effective medium (CEM) method and its application to the epitaxy of metals on metals. Static interaction energy and dynamic Monte Carlo (MC) and molecular dynamics (MD) calculations are employed to examine equilibrium structures and the mechanisms in which equilibrium is obtained. Three specific applications are considered: (1) the mechanism for the formation of the ordered c(2 × 2) Au/Cu(100) surface alloy; (2) The formation of Ag/Rh/Ag(100) “sandwich” structures; (3) The growth of Au on Ag(110).

Corrected effective medium method. V. Simplifications for molecular dynamics and Monte Carlo simulations

We present the conceptual and formal simplifications of the recently developed corrected effective medium (CEM) theory that enable this theory to be used directly in molecular dynamics (MD) and Monte Carlo (MC) simulations of large systems, hence the acronym MD/MC-CEM. The essential idea involves adjustment of the CEM embedding functions to include approximately the original explicit correction for kinetic-exchange-correlation energy differences between the real system and the many atom–jellium systems used as the zeroth order model. Examples of this construction are provided for the Ni, Pd, Ar, and H/Pd(111) systems. Finally, a few brief applications of this method to large systems are provided. These include relaxation of metal surfaces, structure of pure Ni and mixed NiCu clusters, sticking of Cu on Cu(100), and the scattering of Ar from H covered Pd(111).

Molecular dynamics simulations of metal adsorbates on metal surfaces: Rh on Ag(100)

The mechanisms of thin-metal film growth on metal surfaces are topics of considerable scientific and technological interest. We have studied the way in which a deposited metal adlayer behaves as a function of metal substrate temperature. Molecular dynamics simulations are used where the interaction energy and the corresponding forces are generated from the recently developed corrected effective medium (CEM) method. In previous work, the CEM method has been shown to predict the geometric and energetic properties of clean and adsorbate-covered surfaces accurately. In this paper, we present results for the Rh on Ag(100) surface system. We show that Rh atoms penetrate the Ag(100) surface by exchanging with Ag atoms in the surface layer structure. Furthermore, this phenomena is found to depend on both the initial coverage of Rh atoms and the Ag substrate temperature. As the substrate temperature increases, the number of Rh atoms exchanging with Ag atoms in the surface increases. Energetic and dynamical aspects of this system are used to understand the reasons why and also the way in which the exchange takes place.

Ni clusters: Structures and reactivity with D2

We report the results from a theoretical investigation of the structures and reactivity of various isomers of Ni clusters in the size range from 4 to 13 atoms. The geometries of the clusters were optimized using binding energy values calculated by the corrected effective medium (CEM) theory. Two different potential energy surfaces were used to describe the interaction between D2 and the Ni clusters. The first used the form and parameters that were determined in the study of H2 dissociative chemisorption on Ni surfaces, while both used atomic positions appropriate to the clusters, the second used the same form but determined the parameters by comparison to CEM values of the H/Ni13 interaction. Using these PES, we investigated the dissociation dynamics of D2 on NiN (N=4–13) clusters by classical trajectory techniques. We found that: (1) for clusters of size less than Ni9 , the rate constants varied strongly with cluster size; and (2) for all size clusters, the rate constants were very sensitive to different isomers. This isomeric variation of the rate constant is discussed in terms of various structural features in different isomers, a number of which do not have any analogy in the dissociative chemisorption on low Miller index surfaces.

Corrected effective medium method. 3. Application to clusters of magnesium and copper

The authors present a number of new concepts in the recently developed corrected effective medium (CEM) theory. First, the general CEM theory is specialized to the case of an infinite periodic three-dimensional solid. Second, construction of a covalent embedding function of the electron density from diatomic and bulk binding potentials is detailed. On the basis of the smoothness of the function in interpolation between the (low density) diatomic and (high density) bulk limits, we argue that this function is dependent only on the types of atoms in a system and not on the number of atoms in the system (i.e., a universal function of the electron density for particular types of atoms). Applications are made to the description of the structures and energies of homogeneous metal clusters containing from 2 to 201 atoms. Predictions of the equilibrium structures and energies of various high-symmetry structures of Mg{sub 3}, Mg{sub 4}, Cu{sub 3}, Cu{sub 4}, Cu{sub 5}, Cu{sub 13}, and Cu{sub 79} are compared to a number of ab initio and SCF-LSD results. The agreement is quite good even for the small clusters and becomes essentially perfect by Cu{sub 13}. Selected two-dimensional distortions of four-, five-, and seven-atom clusters are considered inmore » detail, with the energetics presented via contour plots. These enable the determination of barriers to isomerization and/or the existence of especially floppy modes of the metal cluster. Geometrical optimization is performed for various configurations of Mg{sub 3-6}, Cu{sub 3-14}, and Cu{sub 19}. These accurate calculations on this range of clusters enable us to investigate the structural stability of the larger metal clusters for the first time. For still larger clusters of up to 201 atoms, we present binding energies for spherical bulk fragments to determine the onset of bulk binding.« less

Corrected effective medium method. II. N-body formulation

A general corrected effective medium (CEM) theory is presented which yields the interaction energy of an N-atom system, in contrast to the previous version of the CEM theory which provides the energy of one atom interacting with the other (N−1) atoms acting as a host. The CEM method presented herein treats all N atoms on an equal basis without identifying all but one as a host, and is referred to by the acronym CEM-N. The basis for this theory involves expressing the interaction energy for the real system in terms of the sum of the interaction energies for each atom embedded into a homogeneous electron gas with compensating positive background (i.e., the effective medium is jellium). Minimization of the difference in kinetic-exchange-correlation energy between the real and effective system, evaluated using density functionals and the approximation of superposition of atomic densities for the system density, yields the prescription for choice of the electron densities of each jellium system. The full interaction energy then consists of three terms: the embedding energy, Coulombic energy, and kinetic-exchange-correlation difference energy. Applications and tests for the C2 , N2, and O2 molecules are presented using the SCF-LD embedding energies of Puska et al. The quality of these results illustrate the need for a new set of universal ‘‘covalent’’ embedding energies, which are constructed semiempirically in the present article.