How to write a mixed number into simplest form

A mixed number is in its simplest form when there is a whole numberand the fraction cannot be reduced any further.

How to write a mixed number into simplest form

Situations Requiring Multiple Patterning[ edit ] There are a number of situations which lead to multiple patterning being required. Sub-Resolution Pitch[ edit ] The most obvious case requiring multiple patterning is when the feature pitch is below the resolution limit of the optical projection system.

Different Parts of Layout Requiring Different Illuminations[ edit ] Different patterns require different illuminations. Different patterns in the same layout as indicated by different colors could require different illuminations, and hence, different exposures.

While horizontal and vertical lines may be addressed with a common illumination, degree orientations would suffer.

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Consequently, to include all these cases would require separate exposures. When patterns include feature sizes near the resolution limit, it is common that different arrangements of such features will require specific illuminations for them to be printed.

The most basic example is horizontal dense lines vs. If both types are used also known as cross-quadrupole C-Quadthe inappropriate dipole degrades the image of the respective line orientation. For example, the array could be exposed with dipole illumination while the periphery could use annular illumination.

The inclusion of both isolated and dense features is a well-known instance of multi-pitch patterning. Subresolution assist features SRAFs have been designed to enable the patterning of isolated features when using illumination tailored for the dense features. However, not all pitch ranges can be covered.

In particular, semi-dense features may not be included.

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Illuminations tailored for certain parts of a multi-pitch pattern may degrade other aspects. Here, the blue locations benefit the minimum line pitch, while the red locations benefit the line breaks but not the minimum line pitch.

how to write a mixed number into simplest form

Sometimes a feature pattern inherently contains more than one pitch, and furthermore, these pitches are incompatible to the extent that no illumination can simultaneously image both pitches satisfactorily. A common example, again from DRAM, is the brick pattern defining the active regions of the array.

Small deviations from 2-beam interference[ edit ] Sidelobe from incomplete cancellation of constructive interference in a broken line pattern. Breaks in such lines, e. Such breaks generally do not dominate the pattern, and are thus small deviations.

These deviations are insufficient to completely offset the constructive or destructive interference of the underlying regular line pattern; sidelobes often result.

Line Cutting[ edit ] Brick pattern image distortion. The difference between cross-line and cross-break image is severe enough that a cut mask is needed. Overlay between line and cut. The cut pattern overlay on the line is important to avoid edge placement error, leaving a portion of the line uncut.

how to write a mixed number into simplest form

Moreover, a dose error can aggravate this. The earliest implementation of multiple patterning involved line cutting. This first occurred for Intel's 45nm node, for nm gate pitch.

Instead it allows definition of features, e. The two-beam interference still dominates the diffraction pattern. The cut pitch can be 1. If the line pitch is already near the resolution limit, the cut pattern itself may have imaging difficulty, from reduced dose or focus window.

Some Definitions

In this case, more than one cut mask would have to be used. However, the use of a cut mask adds the risk of feature damage from an overlay error. Self-aligned cutting to be discussed below may be a preferred option.

Photoresist coating over first pattern; etching adjacent to previous features; Photoresist removal The earliest form of multiple patterning involved simply dividing a pattern into two or three parts, each of which may be processed conventionally, with the entire pattern combined at the end in the final layer.

This is sometimes called pitch splitting, since two features separated by one pitch cannot be imaged, so only skipped features can be imaged at once.

Fractions |

The additional cost of extra exposures was tolerated since only a few critical layers would need them. A more serious concern was the effect of feature-to-feature positioning errors overlay. Consequently, the self-aligned sidewall imaging approach described below has succeeded this approach.

Double patterning by pitch splitting. Double patterning by pitch splitting involves assigning adjacent features to two different masks, indicated by the different colors.

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How do I Find the Simplest Form of a Mixed Number? | Sciencing