Substrate modeling

To enable reflectometry to provide accurate, reproducible, and efficient measurements, several factors must be considered. Choice of the substrate material, substrate modeling, number of measurements per wafer, choice of the measurement patterns, and the setup of the pattern recognition program are all critical to the measurement process, as discussed in the following. 

In reflectometry, the light passes through the films to be measured. Beneath the transparent films, there must be an opaque substrate through which light does not pass. The substrate characteristics must be modeled correctly to calculate the thicknesses of the films above. In silicon processing, theoretically, any of the commonly used metal materials, such as the titanium nitride (TiN), aluminum (Al), and tungsten (W), can be used as substrates. However, in reality, whereas a PMD oxide can be measured on the polysilicon material used in poly interconnections, an ILD oxide can not be measured directly on TiN, because the TiN layer used is too thin to be opaque. TiN is semitransparent if its thickness is less than 1000 A. A thin 

TiN film of approximately 300 A is typically used in the back-end interconnect process, as both the cap layer for the aluminum metal deposition sequence and an antireflective coating for the subsequent photolithography step. Since this TiN cannot be a substrate for oxide thickness measurement in ILD, the aluminum beneath the TiN must be used as the substrate. In other words, the TiN is a component of the film to be measured. Thus, its refractive index or thickness must be known to determine the unknown oxide thickness. However, the refractive index of TiN is not constant, but varies with thickness. As a result, the TiN thickness must be precisely controlled to enable the validity of the substrate modeling. 

Also shown in Fig. 2 are the measured spectrum of a TEOS (tetra-ethyl-ortho-silicate)-TiN-Al stack, overlaid with the theoretical spectrum. The theoretical spectrum was based on such substrate modeling treatment. Given this empirical spectrum, both the TiN and TEOS thicknesses can be simulated and determined simultaneously. Alternatively, the TiN thickness can be fixed with only the TEOS needing to be measured. The latter method is better than the former because the more unknown film thicknesses there are, the more error is introduced. This can be demonstrated by the data in Table I, where the measurement results of the oxide thicknesses on TiN-Al substrates on the same wafer, for TiN thickness fixed and varying, are compared. 

We see from the thickness nonuniformity that if the TiN thickness is fixed, the oxide thickness has a lower standard deviation, indicating a tighter distribution of the measurement results. Since the measurement is on the same wafer, the difference suggests that the effect of fixed TiN thickness to help improve the repeatability of the measurement. (Of course, the TiN film must be uniform for this to be true.) In short, if the control of the TiN 

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The three models used are described by Eq. (6-8) below. The Eqn. (6) is the first-order model based on Michaelis-Menten model, Eqn. (7) is the second-order model, and the Eqn. (8) is the competitive-substrate model. Rso represents the initial specific reaction rate for the substrate S. 

Korzekwa, K.R., Krishnamachary, N., Shou, M., Ogai, A., Parise, R.A., Rettie, A.E., Gonzalez, F.J. and Tracy, T.S. (1998) Evaluation of atypical cytochrome P450 kinetics with two-substrate models evidence that multiple substrates can simultaneously bind to cytochrome P450 active sites. Biochemistry, 37 (12), 4137-4147. 

In a subsequent study, Agrawal, Raff and Thompson showed that the sticking probability for the molecule was independent of the interactions used for the substrate atoms. The mobility of the H atoms and the rate of energy transfer between the H atoms and the substrate, however, were reported to depend somewhat on the lattice. Despite the small dependence on the substrate model, the major results of the initial study remained unchanged. 

N. P. (1997) A refined substrate model for human cytochrome P450 2D6. Chem. Res. Toxicol. 10, 41-48. 

Cyclic mew-configurated 1,2-dicarboxylic acid dimethyl esters are excellent substrates for pig liver esterase90. Cyclopropanedicarboxylales have been studied not only for synthetic reasons, but also so that an active-site and/or substrate model of pig liver may be developed13 5. The results obtained, compounds 11-17, are a good demonstration of the scope and limitation of PLE in asymmetric synthesis. Enantiomeric excesses of the monoesters can be determined by conversion into the amides with (S)-l-phenylethylamine and analysis either by GC or H-NMR spectroscopy, whereas the absolute configuration rests on chemical correlation. 

Box 17.1 Monod-Limiting-Substrate Models of Microbial Population Growth... 

Box 17.1 Monod Limiting-Substrate Models of Microbial Population Growth Case 1 Single limiting substrate, i (Monod, 1949) ... 

The full kinetic scheme for the two-substrate model is given in Figure 3. If product release is fast relative to the oxidation rates, the velocity equation is simplified to Eq. (10) ... 

A second type of nonhyperbolic saturation kinetics became apparent during studies on the metabolism of naproxen to desmethylnaproxen (32). Studies with human liver microsomes showed that naproxen metabolism has biphasic kinetics and is activated by dapsone (T. Tracy, unpublished results). The unactivated data shows what appears to be a typical concentration profile for metabolism by at least two different enzymes. However, a similar biphasic profile was obtained with expressed enzyme (25). This biphasic kinetic profile is observed with the two-substrate model when V/rn2 > Eml and Kml Km2. The appropriate equation for the two-site model when [S] < Kml is... 

Korzekwa KR, Krishnamachary N, Shou M, et al. Evaluation of atypical cytochrome-P450 kinetics with 2-substrate models-evidence that multiple substrates can simultaneously bind to cytochrome-P450 active sites. Biochemistry 1998 37 4137-4147. 

Fassaert et al. (68) simulated H adsorption on a Cu surface by adding an additional electron per metal atom to the system. This approximation relies on the fact that atomic wave functions and energy levels are not too different for Ni and Cu and that their principal difference lies in the number of valence electrons. In the case of adsorption to Cu substrate, which has no unfilled d orbitals, the metal d orbitals do not participate in the bonding to H. All bonding takes place using the metal 4s orbitals. The calculated covalent bond energy is comparable on the Ni and Cu substrate models, so that from the results a distinction between the catalytic properties of the two metals cannot be made. 

The changes in infrared vibration frequency of CO upon adsorption to NiO have been examined by Politzer and Kasten (71) using EH theory. These charge-dependent iterative calculations were used to determine energies of adsorption, and the wave functions were analyzed to determine overlap populations, which were found to be a measure of vibrational frequency. The substrate model consisted of a single Ni or 0 ion with charge appropriate to the system under investigation. 

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