Grove Model Research Articles

Si nanocrystals by ultra-low energy ion implantation for non-volatile memory applications

In nanocrystal (nc) metal-oxide-semiconductor (MOS) memory structures, a fine control of nc location and population is required for pinpointing the optimal device architectures. In this work, we show how to manipulate and control the depth-position, size and surface density of two dimensional (2D) arrays of Si ncs embedded in thin (<10 nm) SiO 2 layers, fabricated by ultra-low energy (typically 1 keV) ion implantation and subsequent annealing. The influence of implantation and annealing conditions on the nc characteristics (e.g. size, density) and the charge storage properties of associated MOS structures is reported with particular emphasis upon the effect of annealing in N 2-diluted–O 2 gas mixture. The latter annealing conditions restore the integrity of the oxide and allow for the fabrication of non-volatile memory devices operating at low-gate voltages. Annealing in diluted oxygen has also an effect on the population of silicon ncs. Their evolution has been studied as a function of the annealing duration under N 2 + O 2 at 900 °C. An extended spherical Deal–Grove model for the self-limiting oxidation of embedded Si ncs has been carried out. It shows that stress effects, due to the deformation of the oxide, slows down the chemical oxidation rate and leads to a self-limiting oxide growth. The model predictions are in agreement with the experimental results.

Si nanocrystals by ultra-low-energy ion beam-synthesis for non-volatile memory applications

In this work, we show how to manipulate two-dimensional arrays of Si NCs in thin (⩽10 nm) SiO 2 layers by ultra-low-energy (⩽1 keV) ion implantation and subsequent thermal annealing. The influence of implantation parameters (dose, energy), annealing conditions (temperature, duration, ambient) and oxide thickness on the NCs characteristics (position, size, density) is reported. Particular emphasis is placed upon post-implantation thermal treatments performed in nitrogen-diluted-oxygen ambient that significantly improve the integrity of the oxide and allow for the fabrication of non-volatile memory devices operating at low-gate voltages. Thermal oxidation in N 2-diluted-O 2 of high-temperature pre-formed silicon NCs has been also examined and modeled using an extended three-dimensional Deal–Grove model. This model reveals that stress effects, due to the deformation of the oxide, slow down the chemical oxidation rate and lead to a self-limiting oxidation of NCs. The model predictions are in agreement with the experimental results.

Kinetics of laser-induced oxidation of silicon near room temperature

The growth of ultra-thin (<6 nm) silicon-dioxide films on Si(100):H, Si(111):H, and a-Si:H surfaces in a dry oxygen atmosphere (0.1–10 Pa) at low temperatures (35–200 °C) was investigated. Oxidation was induced by pulsed F2-laserradiation at 157 nm. The thickness and composition of the growing films were monitored in real time by spectroscopic ellipsometry in the photon energy range of 1.15–4.75 eV. The kinetics of low-temperature oxidation was similar for the Si surfaces investigated and differs from that of high-temperature thermal oxidation (900–1200 °C) that can be described by the Deal–Grove model. To explain the faster growth at the initial stage, it is proposed that oxidation occurs by diffusion of oxygen atoms O and/or ions O-rather than oxygen molecules. The recombination of diffusive species to oxygen molecules limits their penetration into the bulk. A diffusion model is developed for low-temperature oxidation which takes into account the recombination process of the diffusive species. Good agreement between theory and experiment is found. The activation energy of diffusion of the active species was found to be 0.15 eV, in agreement with previous results and recent calculations for O- ions.

Modified Deal Grove model for the thermal oxidation of silicon carbide

A modified Deal Grove model for the oxidation of 4H-SiC is presented, which includes the removal of the carbon species. The model is applied to data on the oxidation rates for the (0001) Si, (0001̄) C, and (112̄0) a faces, which are performed in 1 atm dry oxygen and in the temperature range 950–1150 °C. Analysis within the model provides a physical explanation for the large crystal-face dependent oxidation rates observed.

Formation of ultrathin silicon oxides—modeling and technological constraints

The “ITRS Roadmap” suggests the necessity of constant reduction of dielectric thickness for a number of important applications in ICs technology. The technology of ultrathin layers is, however, very difficult. So is modeling. During the last two decades a number of theoretical models of silicon oxidation, based on different approaches and assumptions, were constructed in order to surpass limitations of the commonly used Deal–Grove model. They will be critically reviewed and discussed in this paper. More attention will be paid to Beck–Majkusiak model, which gives precise predictions even for ultrathin oxide thickness regime, for both: classical oxidation in furnace, and for processing in RTO reactor, and is consistent with description of plasma oxidation process. Some problems of modeling result from technological constraints or measurement methods available. These issues are also briefly addressed in this paper.

Optimum daily ration for Eurasian perch Perca fluviatilis L. reared at its optimum growing temperature

Four feeding experiments were conducted to estimate the optimal daily food ration of Eurasian perch at different stages of its development under optimum rearing temperature (23 °C). Mean initial weights of 0.22-, 0.73-, 1.56- and 18.9-g fish, previously habituated to a dry diet, were used in experiments lasting between 14 and 30 days, depending on initial body weight. Daily feeding levels ranged from 2% to 20% of biomass in 0.22-g fish, from 1% to 8% in 0.73-g fish, from 1% to 10% of biomass in 1.56-g fish and from 0.3% to 3% of biomass for 18.9-g fish. Feeding levels significantly affected survival of perch, except in experiment 4 with large juveniles (18.9). In experiments 2 and 3 (with 0.73- and 1.56-g fish, respectively), cannibalism was significantly higher at the lowest feeding level (1% biomass day −1), while starvation seemed to be the major cause of death in experiment 1 with 0.22-g fish. Values of specific growth rate (SGR) plotted against daily food ration allowed the maintenance, optimum and maximum feeding levels to be determined, according to the Brett and Grove's model. Optimum feeding level for Eurasian perch decreased from 7.4% to 5.1%, 4.5% and 2.2% biomass day −1 for fish of 0.22, 0.73, 1.56 and 18.9 g initial body weight, respectively.

A Review of the Role of Excess si in SIO2 at the Growing Oxide Interface.

AbstractThis paper discusses the issues that arise when ab excess of Si or SiO flux is used to adjust the Deal -Grove model to match the observed growth in the early stages of oxidation. The author uses the insight gained from the modeling of the oxidation of Si-Ge alloys as a background to the discussion. A recent simulation of the thin oxide problem by M. Uematsu et.al. invokes the role of “excess Si” near the interface to adjust the D&amp;G reaction coefficient and also add additional flux to the growing oxide.This paper assesses some of the points in this model as well as the relation with sub bonded Si and/or the reaction barrier layer to oxidationmechanism. A recent proposal by Pasquarello et.al opens new insight into the way the interface may adjust to the growing oxide which challenges our previous mechanistic notions.

Interface structure between silicon and its oxide by first-principles molecular dynamics

The requirement for increasingly thin (<50 A) insulating oxide layers in silicon-based electronic devices highlights the importance of characterizing the Si–SiO2 interface structure at the atomic scale. Such a characterization relies to a large extent on an understanding of the atomic-scale mechanisms that govern the oxidation process. The widely used Deal–Grove model invokes a two-step process in which oxygen first diffuses through the amorphous oxide network before attacking the silicon substrate, resulting in the formation of new oxide at the buried interface1. But it remains unclear how such a process can yield the observed near-perfect interface2,3,4,5,6,7,8,9,10,11,12. Here we use first-principles molecular dynamics13,14,15 to generate a model interface structure by simulating the oxidation of three silicon layers. The resulting structure reveals an unexpected excess of silicon atoms at the interface, yet shows no bonding defects. Changes in the bonding network near the interface occur during the simulation via transient exchange events wherein oxygen atoms are momentarily bonded to three silicon atoms — this mechanism enables the interface to evolve without leaving dangling bonds.

Perturbation solution for the interfacial oxidation of silicon

A new perturbation solution for the oxidation of silicon that properly accounts for diffusion of molecular oxygen and reaction kinetics at the silicon–silicon dioxide interface is presented. The perturbation treatment is based on the assumption of a dilute solution of O2 in SiO2 phase. We show that the solution obtained by Deal and Grove [J. Appl. Phys. 36, 3770 (1965)] is the zeroth perturbation solution and formally prove that it is a quasi-steady-state solution. In the limiting case of instantaneous reaction, the perturbation solution is shown to be equivalent to the result obtained by Peng, Wang, and Slattery [J. Vac. Sci. Technol B 14, 3316 (1996)]. The proposed perturbation scheme is developed here for first-order reaction kinetics, but can easily be extended to other rate expressions. The higher order corrections offered to Deal and Grove model predictions might provide a needed method for assessing the error incurred when these predictions are used to estimate oxide film thickness in practical situations.

Diffusion of near surface defects during the thermal oxidation of silicon

The diffusion of defects during the thermal growth of SiO2 film on Si(100) in dry O2 was investigated using sequential treatments in natural oxygen (16O2) and in heavy oxygen (18O2) in a Joule effect furnace. The O18 depth profiles were measured with a depth resolution better than 1 nm, using the nuclear reaction narrow resonance O18(p,α)15N (ER=151 keV, ΓR=100 eV). From these profiles, we confirmed that just below the surface an exchange between the oxygen atoms from the gas phase and those from the silica occurs, even for silica films thicker than 20 nm. This fact is not predicted by the Deal and Grove model. A diffusion of oxygen related defects takes place in the near surface region, with an apparent diffusion coefficient D*=4.33×10−19 cm2/s for an oxidation temperature of T=930 °C and for an oxygen pressure of P=100 mbar.

Growth law of silicon oxides by dry oxidation

A theoretical description of the kinetic mechanism of thermal oxidation in silicon is proposed by complementing the Deal - Grove model. The relationship of the classical linear - parabolic growth law is generalized to the logarithmic growth law which provides a complete description for the whole regime of oxide films. In particular, the enhanced oxidation rate in the thin regime may be attributed to the diffusion length, which is characterized by the difference between the activation energies of the diffusion process and the reaction process. Our fitting of the logarithmic growth law to several experimental results shows excellent agreement and the fitting parameters also provide activation energies of 1.50 and 2.49 eV for the interfacial reaction and diffusion in oxide respectively.

An isotopic labeling study of the growth of thin oxide films on Si(100)

The mechanism of thin (&amp;lt;8 nm) oxide growth on Si(100) has been studied by high-resolution medium energy ion scattering in combination with oxygen isotope substitution in the T=800–900 °C and 0.1–1 Torr oxygen pressure regime. Isotopic labeling experiments demonstrate that the Deal–Grove model breaks down for these films. In addition to the traditional oxidation reaction at the Si/SiO2 interface, two other spatially specific reactions take place during thermal oxidation: an exchange reaction at the oxide surface and an oxidation reaction in the near-interfacial region.

Oxidation Kinetics of Chemically Vapor‐Deposited Silicon Carbide in Wet Oxygen

The oxidation kinetics of chemically vapor‐deposited SiC in dry oxygen and wet oxygen (PH2O= 0.1 atm) at temperatures between 1200° and 1400°C were monitored using thermogravimetric analysis. It was found that in a clean environment, 10% water vapor enhanced the oxidation kinetics of SiC only very slightly compared to rates found in dry oxygen. Oxidation kinetics were examined in terms of the Deal and Grove model for oxidation of silicon. It was found that in an environment containing even small amounts of impurities, such as high‐purity Al2O3 reaction tubes containing 200 ppm Na, water vapor enhanced the transport of these impurities to the oxidation sample. Oxidation rates increased under these conditions presumably because of the formation of less protective sodium alumino‐silicate scales.

Oxidation of silicon in plasma afterglows: New model of oxide growth including recombination of diffusing O atoms

The mechanism of plasma afterglow oxidation of silicon by atomic oxygen is discussed in terms of a physical model that includes recombination of the oxidant atoms during their diffusion through the SiO2 layer. Inclusion of a first-order O loss term in the continuity equation that governs the unbound O atoms leads to a biexponential concentration profile in the oxide. The corresponding time-dependent O flux across the SiO2/Si interface results in an oxide growth equation that is a more general form of the classical Deal–Grove model. Confrontation with available experimental data shows that the general expression can be abbreviated for oxide widths w≥0.1 nm as t= (A2/B)[exp(w/A)+exp(−w/A)−exp(wi/A)−exp(−wi/A)], where wi is the native oxide width. The two model parameters are A≡√D/k and B≡2HD[O]g/nb, with D being the diffusivity and k the first-order loss rate constant of unbound O atoms in SiO2, H the SiO2/gas Henry equilibrium ratio of free O atoms, [O]g=the gas-phase O atom concentration, and nb=the bound-O number density in SiO2. The two-parameter model provides excellent fits (σ≂2% of final w) to the available data on both n- and p-type Si, strikingly better than fits obtained by the Deal–Grove equation in particular for p-type Si. The values deduced for the model parameters A and B provide proof for the controlling importance of the O-atom recombination process, especially for p-type Si. The model parameters also allow values to be derived for other pertaining physical constants; e.g., the product HD is deduced to be 2.4×10−9 cm2 s−1 at T≂850 K, in close agreement with the known HD value for neon atoms in SiO2, equal to 5.2×10−9 cm2 s−1.

A note on the linear-parabolic law of phase formation

Abstract One of the two equations (reaction rate) on which the Deal—Grove model of Si oxidation is based, assumes that the reacting Si is in thermodynamic equilibrium. The diffusion of injected interstitials, necessarily along a chemical potential gradient, provides evidence of an increase of the chemical potential of the substrate atoms at the interface. This must be accounted for in any kinetic equation of oxide formation and growth. A first approximation model based on this observation leads to a modified linear—parabolic equation which gives an improved match between the calculated growth curve and experimental data. The physico-chemical values that can be extracted from the model are compatible with commonly accepted ideas about the properties of silicon and of SiO2. The approach is deliberately general and is thought to be valid not only for the oxidation of silicon, but also for the oxidation of metals and for all solid-state chemical reactions. The model derived should be valid for many solid phas...

Oxidation property of silicon small particles

The oxidation rate of silicon small particles prepared by a gas-evaporation method and with diameters ranging from about 20 to 500 nm is investigated by high-resolution transmission electron microscopy. It is found that the oxidation rate of silicon small particles is less than the rate for wafers, and decreases with decreasing particle size. This phenomenon cannot be explained by the Deal–Grove model.

Fluorine-enhanced thermal oxidation of silicon in the presence of NF3

The oxidation rate of silicon has been dramatically increased at temperatures between 600 and 800 °C by adding NF3 gas to a dry oxygen atmosphere. The chemical analysis of the resulting oxide has revealed that fluorine atoms in SiO2 are incorporated as Si–F bonds. The existence of such bonded-fluorine atoms enhances the diffusion of oxidant through the oxide layer. The oxidation reaction rate in the SiO2-Si interface is also remarkably increased because fluorine atoms in the interface create silicon dangling bonds and weakly positively charged silicon atoms as well. The kinetics of the fluorine-enhanced oxidation has primarily been interpreted in terms of the Deal–Grove model.

An improved high-frequency MOS capacitance formula

Minority carriers within the inversion layer of an inverted MOS capacitor can move in response to a high-frequency ac signal even though the total minority-carrier charge is fixed at the value set by the dc bias. In this paper a simple explicit formula for the high-frequency capacitance which includes this ac inversion layer polarization effect is derived. The derivation is based upon the assumption that the distribution of minority carriers within the inversion layer is governed by a constant quasi-Fermi level, independent of position within the inversion layer. Hence, the resulting capacitance formula is restricted to the case where minority-carrier trapping by defects in the inversion layer or at the interface do not significantly hinder the ac response of the minority carriers. Comparison of the new formula with the calculations of Sah, Pierret, and Tole, which have as their main approximation the neglect of the ac minority-carrier movement, shows that the neglect of this effect leads to a 7% error in strong inversion at 1013/cm3. Comparison with the commonly used formulas of Lindner and of Grove, which contain approximations in addition to the neglect of inversion layer polarization, shows that these formulas are in error by 9% and 4%, respectively, in strong inversion at a doping level of 1013/cm3. Although more accurate than these other calculations in strong inversion, the Grove model is less accurate in depletion and weak inversion. Figures illustrating the error involved in all three earlier approaches as a function of bias with doping level as a parameter are presented.

The Bierwag and Grove Model of the Term Structure of Interest Rates: An Alternative British Test

Bierwag and Grove [ 1 ] have recently presented an interesting model of the term structure of interest rates which is analytically more appealing than the Meiselman model [5]. In particular, they are able to dispense with the assumption of identical singlevalued expectations and show that individual wealth holders seeking to maximize utility will determine an equilibrium forward rate which is a weighted average of the individual predicted rates. The original Meiselman model is incorporated into this model as a special case. Tests of these new models give good results for the United States using the Durand data [3] but for Britain, using the Grant data [4], the results are generally very poor. This note shows that an alternative set of annual British data, covering yields on government securities for 1933 to 1963, gives results which compare favorably with those for the United States and it is not necessary to conclude, as do Bierwag and Grove, that the expectations mechanism is different in the United Kingdom. The improved results also illustrate the point, elaborated in [2], that the method of yield estimation is the critical factor in tests employing forward interest rates derived from an estimated yield structure. The theoretical model of the term structure developed by Bierwag and Grove shows how a given investment fund is allocated among short and long term bonds given the investor's utility function and the first two moments of his probability distribution of expected interest rates.' The market equilibrium forward rate is then shown to be a weighted average of individual predicted rates. In order to make the model operational, empirical specifications for the formation of interest rate expectations are required. These specifications are of two types: a traditional adaptive expectations function and a Meiselman type adaptive function.2 The relevant equations are specified for each type before the empirical results of the British test are reported.3