Preston Coefficient

Preston coefficients may be readily measured for a polish process. The simplest method for measuring is to measure the polish rate and divide by the pressure and velocity. However, this method is sensitive to error in the polish rate measurement. A better measure of is obtained by measuring the polish rate over a range of velocities and pressures. If the polish rate behavior follows Preston s equation, i.e., if it is linear and intersects the origin, may then be obtained by differentiating Equation (4.1) with respect to either pressure or velocity. is then given by  

the slope of a plot of the polish rate vs. either pressure or velocity gives K. In the next section, polish rate curves for copper are given. From these curves, the Kp values calculated are compared to the theoretical K values prc cted by Equation (4.2). 

To maintain consistent removal rates, pad conditioning was employed in these experiments. Removal of the used slurry was performed using a razor blade held against the pad with light force and drawn from the center to the edge of the pad while a fresh so- 

Copper polish rate vs. pressure on the Strasbaugh tool with Suba IV pad using 125 mtn wafers, a slurry of 2.5 wt% AI2O3 and 2.0 vol% NH4OH, a slurry flow rate of 250 ml/min, and a velocity of 130 cm/sec. 

---

A number of additional wafer-level variations in the removal rate may result from the variation in the Preston coefficient across the surface of the wafer. Because depends on the slurry, pad, material, and process parameters in a poorly understood fashion, it is not clear where all of these dependencies ultimately reside. 

Considering only the mechanical properties of the film being polished, Brown et al/ give an expression for the Preston coefficient ... 

According to the Preston equation (Equation (4.1)), the polish rate varies linearly with pressure and velocity. In general, the Preston equation describes the pressure and velocity dependence of oxide CMP rate well, as shown in Figure 5.14. However, the theoretical value of the Preston coefficient, = 1/2E, does not explain the polish rate variation with other important process variables such as pad type, pad condition, slurry abrasive, and slurry chemicals. 

Table 7.3 lists the calculated Preston coefficients for a given polishing condition. Comment on the difference between the calculated (Kp = 1/2 E) and experimental values for Cu. 

Figure 2.24 shows the removal rate variation as a function of pattern density for five dies with the positions shown in Figure 2.23a. This trend indicates that the removal rate decreases linearly as the pattern density increases. It can intuitively be explained by Preston s equation, R = k pv, where k, is the Preston coefficient, R is the material removal rate, p is pressure, and V is relative velocity. As pattern density increases, the effective...

The term ATm can be modeled following the Preston formalism (Rock et al., 2011) = PvKp, where P and v are the down pressure and the platen speed of CMP, respectively. ATp is an effective Preston coefficient. Kp = (mEm)/Bmc. with Bmc and Tm denoting the molar binding energy and molar volume of MC, respectively fi is the effective coefficient of friction of the pad—metal interface. In this description of chemically dominated CMP, one has MRR = 0 if the surface complex does not form. If no surface complexes or soluble species are formed, and if mechanical abrasion of the unmodified metal surface is the only means of material removal, then Eqn... 

Since the force transmissions take place only over the asperity contact spots, the macroscale Preston s law RR = K V requires some careful modifications. Usually, the Preston coefficient is taken as a constant containing all relevant effective material properties for polishing between two flat smfaces. For polishing in the contact spot between an asperity and the wafer, Vasilev (Vasilev et al., 2011) proposed a microscopic formulation of Preston s law. That is, the removal caused by one contacting asperity is calculated as... 

For polyelectrolytes the second virial coefficient is very sensitive to ionic strength. Preston and Wik [28] have shown a tenfold increase in B—from 50 mlmolg" to 500 ml mol g —upon decreasing the ionic strength from 0.2 down to 0.01 moll h... 

Figure 9.6 Calculated internal conversion coefficients for (a) electric transitions and (b) magnetic transitions. (From M. A. Preston, 1962, p. 307.) Copyright 1962 by Addison-Wesley Publishing Company. Reprinted by permission of Pearson Education.

Fig. 1. Plot of the logarithm of calculated partition coefficients versus the logarithm of the capacity factors for tyrosine (1), 3-iodo-tyrosine (2), and 3,5-diiodotyrosine (3). The capacity factors were measured on a -Bondapak Cig column with a mobile phase of 10 % methanol-water-5 mM orthophosphoric acid, adjusted to pH 3.0. Reprinted with permission from Hearn et eil. (10). Copyright by Preston Publications Inc., Niles, Illinois.

Analyses of Stribeck curves, Preston s coefficient, COF, and tribological mechanism indicator correlating with each other help to understand the polishing mechanisms. Such an analysis not only helps in the process development but also provides useful feedback to the pad development manufacturers. 

Assuming the Preston s coefficient is given as 1/2 E where E is the Young s modulus of the film, tabulate the coefficients for Si, SiOj, Al, Cu, polysilicon, W, parylene, and a photoresist of your choice. 

For metal polish, chemical effects can be explicitly delineated from Preston s coefficient Eq. (1) ... 

Little work has been done to compare the nature of ligands in riverine, estuarine, and coastal waters. Preston (1979) found similar selectivity coefficients for copper with humic compounds isolated from different salinity regimes of the Tamar estuary. His results are made uncertain by lack of knowledge of the molecular weights of the compounds, but it appeared that the selectivity for copper decreased with increasing salinity. The stability constant data of Mantoura et al. (1978) also show similar selectivities for copper by aquatic humic substances from river, lake, and marine waters, which would imply that little variation in selectivities should be found along an estuarine salinity gradient. 

The dramatic influence of even small proportions of a r tes on the results from q.Ls. has been clearly demonstrated by, for example, Preston and coworkers in related studies on proteoglycans. Another difficulty that is often not reported is the contribution to error caused by concentration measurement (of the unsolvated solute) concentrations can rarely be measured to better than 5%, and will contribute error in both the Zimm plot and the values for the refractive increment used for evaluation of the constant If q.Ls. and the Svedbeig equation are used, errors in concentration will also be manifested in the extrapolations of the diffusion and sedimentation coefficients. 

As in vapor-liquid equilibria and in liquid-liquid equilibria, exparimental arixhire data are required to find liquid-phase activity coefficients. For some so]id-liquid systems, estimates for 7 can be mede using regular-solution theory (Preston and Prausnitz2) or UNIFAC (Cmehling el al,3), but for reliable work a few exparimental measurements are necessary. 

Equation (1.6-8) has been applied by Snave to correlate solubilitses of solid carbon dioxide in liquid hydrocarbons at low temperatures. Alternatively, My res and Prausnitz. Preston and Prausnitz,3 and more tacantly Teller and Knapp6 also coreslated such solubilities but they ured the conventional activity-coefficient method Eqs, (1,6-3) and (1.6-6). 

The internal conversion coefficient for Fe has been accurately determined from the transmission integral (seeO Eqs. (25.11), (O 25.30), and (O 25.31)) by using a series of iron absorbers all having the same thickness of iron atoms but the enrichment in Fe varied by a factor of as much as 37 (Hanna and Preston 1965). 

---