Sources of Nonuniformity

There are several sources of nonuniformity that might give rise to the singular perturbation expansions. Some of these are the following  

Type change of a partial differential equation (e.g., from parabolic to hyperbolic equation). 

Examples of the nonuniformity source for infinite domain come from problems involving cos(r) and sin(0 functions, and the higher order terms involve t sin(0 or t cos(f), which would make the higher order terms more secular (i.e., more unbounded) than the preceding term. 

The second source of nonuniformity may seem obvious. For example, when the highest derivative is removed (i.e., when the small parameter is set to zero) the differential equation is one order less, and hence, one boundary condition becomes redundant. To invoke this boundary condition, there must exist a boundary layer wherein a boundary condition becomes redundant when the parameter s is set to zero. 

To demonstrate sources of nonuniformity, let us study the following simple second order differential equation with a small parameter multiplying the second order derivative 

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In Section II, we focus first on wafer-scale models, including macroscopic or bulk polish models (e.g., via Preston s equation), as well as mechanistic and empirical approaches to model wafer-scale dependencies and sources of nonuniformity. In Section III, we turn to patterned wafer CMP modeling and discuss the pattern-dependent issues that have been examined we also discuss early work on feature-scale modeling. In Section IV, we focus on die-scale modeling efforts and issues in the context of dielectric planarization. In Section V, we examine issues in modeling pattern-dependent issues in metal polishing. Summary comments on the status and application of CMP modeling are offered in Section VI. 

More complicated methods have been developed for nonuniformity compensation. For example, AMOLED displays have also been demonstrated with a photodetector in each pixel to sense OLED hght output and apply feedback to correct for nonuniformity [55], It should be noted that this is the only approach that compensates for OLED luminance decay over operational hfe, since none of the others detects light output. However, the method introduces new sources of nonuniformity arising from the photodetector and its associated circuitry, and it is unclear whether the net effect will always improve display uniformity. 

Over the years, users of perturbation methods have evolved a shorthand language to express ideas. This reduces repetition and allows compact illustration. We first present the gauge functions, which are used to compare the size of functions, and then we present the order concept, which is convenient in expressing the order of a function (i.e., the speed it moves when e tends small). Finally, we discuss asymptotic expansions and sequences, and the sources of nonuniformity, which cause the solution for 0 to behave differently from the base case. 

Another important factor for coated structures is the presence of defects in the protective coating. Not only does the size of a defect affect the current but also the position of the defect relative to the anode. Current tends to be concentrated locally at defects. A fundamental source of nonuniformly distributed CP current over structures results from an effect known as attenuation. In long structures such as pipelines the electrical resistance of the structure itself becomes significant. The resistance of the structure causes the current to decrease nonlinearly as a function of distance from a drain point. A drain point refers to the point on the structure where its electrical connection to the anode is made. This characteristic decrease in current (and also in potential), shown in Fig. 11.11, occurs even under the following idealized conditions ... 

Basic box models cannot portray effects of nonuniform source patterns. If, for particular chemical species or particular source classes, the dependence of emissions on population density or other identifiable parameters is apparent and significant, we have used modifications to the box modeling approach. As an example, it might be assumed in modeling products of combustion of the lighter fuel oil distillates that source distribution patterns are proportional to population density patterns, because most of such fuel is burned in residential furnaces in cold weather cities. 

Sources of Wafer-Scale Nonuniformity a. Relative Velocity Mismatch Across the Wafer... 

Optimizing the fresh catalyst physical properties including particle density, PSD, and attrition resistance is critical to maintaining acceptable fluidization and resulting circulation of the catalyst inventory. Excessive attrition of the catalyst will lead to nonuniform fluidization and disrupt circulation. Potential sources of attrition include ... 

A few remarks are due about this feature. The nonuniformity above is a formal expression of breakdown of the local electro-neutrality assumption in concentration polarization, described in the previous chapter. Essentially, this reflects the failure of a description based upon assuming the split of the physical region into a locally electro-neutral domain and an equilibrium double layer where all of the space charge is concentrated. The source of this failure, reflected in the nonuniformity of the corresponding matched asymptotic expansions, is that the local Debye length at the interface tends to infinity as the voltage increases. In parallel a whole new type of phenomena arises, which is not reflected in the simplistic picture above. The... 

In this illustration, the spatially nonuniform body forces cause an internal source of vortic-ity. Vorticity is also generated due to shearing behavior at the walls. 

Radiation Dose Limits. For routine exposure of individual members of the public to all man-made sources of radiation combined (i.e., excluding exposures due to natural background, indoor radon, and deliberate medical practices), NCRP currently recommends that the annual effective dose should not exceed 1 mSv for continuous or frequent exposure or 5 mSv for infrequent exposure. The quantity effective dose is a weighted sum of equivalent doses to specified organs and tissues (ICRP, 1991), which is intended to be proportional to the probability of a stochastic response for any uniform or nonuniform irradiations of the body (see Section 3.2.2.3.3). 

The engineering objectives of die design are to achieve the desired shape within set limits of dimensional uniformity and at the highest possible production rate. This chapter discusses both objectives, but the question of die-formed product uniformity deserves immediate amplification. To understand the problem, we must distinguish between two types of die-formed product nonuniformity (a) nonuniformity of product in the machine direction, direction z [Fig. 12.2(a)], and (b) nonuniformity of product in the cross-machine direction, direction x [Fig. 12.2(b)]. These dimensional nonuniformities generally originate from entirely different sources. The main source of the former is the variation... 

A hot fluid model would be highly desirable for applications in astrophysics. As we have already mentioned, the formation of RES in the primordial plasma could be an important source of large-scale nonuniformities in density and temperature, which seeded the formation of galaxies and clusters of galaxies [4], In particular, it is conjectured that in the early universe matter was present in the form of a mixture of electrons, positrons and photons in thermal equilibrium at a temperature above me2. It is evident that the propagation of relativistic EM waves in such peculiar environment should be addressed in the framework of a hot-plasma model. 

To properly understand CMP film thickness control, the CMP engineer should understand the sources of thickness variation and how they impact the total film thickness uniformity. Nonuniformity can be grouped in two categories—random variation and systematic variation. Examples of random variation include wafer-to-wafer (WTW), run-to-run (RTR), and some elements of within-wafer (WIW) variations. Elements of random variation add to the total thickness variation by their root mean square [19]... 

All the just-mentioned analyses focus more or less on fluidized beds and under unbounded conditions. The analysis of all these investigators concentrated on whether a periodic disturbance in the axial direction grows with respect to time, leading to instability, or decays with time, indicating a stable system. The treatment is strictly for axial direction and the axial nonuniformities are the source of transition. Analyses such as these have the limitation that the real fluidized bed is not of infinite extent, and that some account must be taken of the boundaries at the upper and lower surface of a bed, the finite depth, and also the walls that bound the beds laterally. Further, the one-dimensional unbounded description does not generally explain all the experimental observations at transition. Shnip et al (1992) have used the theory of hnear stabihty for the analysis of bounded beds. 

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