Handbook of algorithms for physical design automation part 73 docx

It can be shown that, when the diffraction pattern of Equation 35.10 is placed at exactly the focaldistance in front of the lens, the field at the focal plane a distance f behind the len

It can be shown that, when the diffraction pattern of Equation 35.10 is placed at exactly the focal

distance in front of the lens, the field at the focal plane (a distance f behind the lens) allows this

phase factor to cancel the phase factor in the Fraunhofer diffraction formula, and the field in the image plane becomes

E (p, q) ∝ −i 1

λf

∞



−∞

∞



−∞

M (x, y)e −i(2π/λ)(xp+yq) dx dy (35.12)

This form will be recognized as a mathematical representation that corresponds to the 2D Fourier transform [30] of the mask pattern:

To actually form the image of the mask at position (x1, y1), the lens aperture and behavior,

represented by a pupil function designated as P (a, b), are multiplied with the diffraction pattern at

the focal point This image in the focal plane is in turn transformed by a second lens at a distance f :

E (x1, y1) ∝ FTP (a, b) · E(p · q)= FTP (a, b) · FTM (x, y) (35.14)

where P represents the pupil function, encompassing the wavefront transforming behavior of the

lens This is illustrated in Figure 35.5

Pupil functions can be simple mathematical structures, such as

P (a, b) =



1, a2+ b2≤ r

0, a2+ b2> r



=



1, ρ ≤ r

0, ρ > r



(35.15)

representing the physical cutoff of the circular lens housing or radius r (shown in both Cartesian (a, b)

and polar coordinates (ρ, φ)) However, additional phase behavior of the lens can also be included

in the pupil function Lens aberrations can be represented by an orthonormal set of polynomials called Zernike polynomials, each representing a specific aberration [29] The Zernike polynomials are generally represented in polar coordinates, following the form

Z j (ρ, φ) = a m

n R n m (ρ)Y m

Table 35.1 below shows a few of the Zernike polynomials and the corresponding aberration More detail on these functions can be found in Ref [29]

Mask

Image System pupil plane

FIGURE 35.5 Simplified representation of the optical system of an imaging tool At the pupil plane, the

amplitude of the field represents a two-dimensional Fourier transform of the object, multiplied with the pupil function

TABLE 35.1

First Ten Zernike Polynomials, an Orthogonal Set of Functions

That Describe the Lens Aberrations

n (φ) Aberration

8 3ρ3− 2ρ Cosφ (Balanced) x-coma

Note: More details can be found in Ref [29].

At this point, the image can be calculated, but the representation is still in terms of the amplitude and phase of the local electric field Photosensors, whether they be the retinas of the eye, a photo-electric cell, or the molecules of a photoresist, produce a signal in proportion to the amount of energy

in the electromagnetic field The energy is proportional to the image intensity, found by squaring the modulus of the electric field:

where∗denotes the complex conjugate operation

35.2.2.3 Linearity

Although actual imaging systems comprise more than two simple phase front transformations, a key theorem on which all lens design is based is that any complex lens can be reduced to a simple Fourier transform, a Pupil function, and an inverse Fourier transform

This is a very powerful result, and is the basis of the entire field of Fourier optics [30] Regard-less of the exact lens structure and configuration, image simulation becomes a simple matter of designating the appropriate coordinate system, computing Fourier transforms and finding the proper

representation of the pupil function P Because Fourier transforms themselves are linear, the optical

system is modeled by a linear process This means that any arbitrary image can be assembled by creating a superposition of images from a suitable set of building blocks, each computed on its own The linearity of the Fourier transform allows a complex 2D pattern to be decomposed into

a Fourier series expansion of different 2D spatial frequencies, each being treated in turn and the final fields summed together Note also that a nonmonochromatic distribution of wavelengths λ

can similarly be computed wavelength by wavelength, and the final results summed as appropriate This linearity holds as long as the media can be adequately descried by a refractive index, as in Equation 35.8

Note that for some materials, optical properties can change in the presence of strong electric fields, and the refractive index itself becomes an expansion:

Materials in which these effects are significant are called ‘nonlinear optical materials’ [31] Clearly, these nonlinearities can cause additional complications if they were to be used in imaging

applications However, values of n2are generally very small, even for highly nonlinear materials, and these nonlinear effects are generally only observed using lasers with extremely high power densities

In general, the assumption that a total E field can be represented by a linear superposition of E fields

remains valid

35.2.2.4 Computation by Superposition

The mathematics of Equation 35.14 represent that the imaging of any particular mask function is the multiplication of the FT of the mask with the pupil function Because multiplication in Fourier

space corresponds to convolution in position (x, y) space, image simulation reduces to the ability to

do the following computational tasks in various combinations:

1 Digitize the mask function into a 2D amplitude and phase pixel array [M (x, y)]

2 Estimate a discrete 2D representation of the pupil function [P (ω x,ω y )]

3 Perform array multiplication (e.g.,[P] • [M] in frequency space)

4 Compute discrete Fourier transforms (and inverse transforms as well)

35.2.2.4.1 Pixel Representation of the Mask

Creating a pixel representation of the mask is usually fairly straightforward Mask layouts are gener-ated using polygons, often exclusively with Manhattan geometries The ability to create an accurate discrete representation of the layout then becomes a question of the resolution desired and the size

of array that can be computationally managed This selection of the address grid can impact the computation and data management properties significantly, so should be done with care Generally,

a grid around 1 nm is selected for contemporary ICs with features as small as 45 nm

35.2.2.4.2 Pixel Representation of the Pupil

Once the pupil function is known, a similar mapping onto a grid is carried out Here, the resolution of the pupil components need not be nearly as dense as the grid selected for the layout However, because the transform of the mask and the pupil must be entry-wise multiplied, some care should be taken to ensure that the two grids match well Although the simplest pupil functions are mathematically easy

to represent (e.g., a circular aperture), these functions do not map to a Manhattan grid in the same way most mask functions can In addition to this, the lens aberrations, also incorporated into the pupil, typically have circular symmetry (Table 35.1) Staircasing of these non-Manhattan functions occurs, and without a very fine grid, the results are less accurate

35.2.2.4.3 Array Multiplication

This is one of the basic computing operations, and is typically straightforward The matrix multipli-cation occurs pixel by pixel, and the entries in the corresponding matrices are therefore multiplied entrywise

⎡

⎢

⎣

P1,1M1,1 P1,2M1,2 P1,3M1,3 P1,4M1,4

P2,1M2,1 P2,2M2,2 P2,3M2,3 P2,4M2,4

P3,1M3,1 P3,2M3,2 P3,3M3,3 P3,4M3,4

P4,1M4,1 P4,2M4,2 P4,3M4,3 P4,4M4,4

⎤

⎥

⎦

=

⎡

⎢

⎣

P1,1 P1,2 P1,3 P1,4

P2,1 P2,2 P2,3 P2,4

P3,1 P3,2 P3,3 P3,4

⎤

⎥

⎦ •

⎡

⎢

⎣

M1,1 M1,2 M1,3 M1,4

M2,1 M2,2 M2,3 M2,4

M3,1 M3,2 M3,3 M3,4

⎤

⎥

where M1,1represents, for example, a pixel of the Fourier transform of M (x, y),

M1,1= FTM (x, y)Pixel(a1,b1) (35.20)

P1,1represents the pupil function at pixel (a1, b1), etc

It is clear from this that the grids of the mask function, the pupil function, and the final image need to be matched to avoid excessive interpolation

35.2.2.4.4 Fast Fourier Transform

The fast Fourier transform (FFT) is one of the best known and widely used computational algorithms [32–34] Normally, a discrete Fourier transform (DFT) numerically executing the Fourier transform

in a brute force manner, would require O (N2) arithmetic operations However, when the functions

to be transformed can be discretized into elements that are a multiple of 2, the DFT can be broken

down into a number of smaller DFTs The final result can be constructed to only have O (N log N)

arithmetic operations In a similar fashion, 2D discrete Fourier transforms can be broken down into

a collection of 1D DFTs, each with a similar gain in computational efficiency

Because the mask function M is well behaved (with values of either 0 or 1, depending on the coordinates) and the pupil function P is continuous, both the mask function and pupil function can

be digitized into a 2D array of pixels, with the number of pixels on each side being some multiple

of 2 The FFT can therefore be used for this computation, and it has become the main engine of image simulation

35.2.3 RET TOOLS

The ability to simulate images quickly with tools such as the FFT and to compose arbitrary images based on the superposition of partial images gives rise to the possibility of EDA tools with dual, com-plementary capabilities: a database engine, to manage and process layout polygons, and a process simulation engine The process engine calls on certain layers of data representing portions of the IC layout, transforms them to simulate processing behavior, and returns a representation of the trans-formed data to the database for further analysis This is illustrated schematically in Figure 35.6 This combination of data management and simulation is how the entire class of ‘resolution enhancement techniques (RETs) [35] are implemented in an EDA flow

Layer 2: Isolation

Layer 3: Gate

Layer 4: Contact

Layer 5: Metal 1

Layer 6: Via 1

Layer 7: Metal 2

Layer 8: Via 2

Other simulated layers

Simulation

Layout

…

FIGURE 35.6 Lithographic simulation of a single layer of an IC layout.

(a) (b)

FIGURE 35.7 Iso-dense bias (a) represents the drawn layout, while (b) illustrates the result on the wafer.

For this process, the lines in the dense region are thinner than isolated lines with the same nominal dimension

(Reproduced from Schellenberg, F.M., Zhang, H., and Morrow, J., Optical Microlithography XI, Proceedings

of SPIE, 3334, 892, 1998 With permission.)

There are three major RETs in use today: Optical and process correction (OPC), phase-shifting masks (PSM), and off-axis illumination (OAI) [35–37] Each corresponds to control and manipulation

of one of the independent variables of the optical wave at the mask: amplitude (OPC), phase (PSM), and direction (OAI) The changes required for OPC and PSM are implemented by changing the layout of the photomask, while OAI is implemented by changing the pattern of light emerging from the illuminator as it falls on the mask

35.2.3.1 OPC

The acronym ‘OPC’, which is now used as a general term for changing the layout to compensate for process effects (optical and process correction), originally stood for optical proximity correction, and was used to predict and compensate for one-dimensional proximity effects One example of a 1D effect, ‘iso-dense bias’ [38,39], is illustrated in Figure 35.7 [40] Here, isolated and dense features

of identical dimension on the photomask print at different dimensions on the wafer, depending on the proximity to nearby neighbors Shown in the ‘pitch curve’ of Figure 35.8 is the characteristic behavior observed for 1D periodic features in a typical optical lithography process [41] In this case,

‘pitch’ is the 1D sum of line and space dimensions

Some of this can be readily understood as an interaction of the Fourier spectrum of the photomask layout and the low-pass properties of the stepper lens and process: Dense lines have a well-defined

Target linewidth

Isolated Dense

Pitch

FIGURE 35.8 Iso-dense pitch curve, quantifying the linewidth changes for nominally identical features (i.e.,

lines all at a single target dimension) as a function of pitch (Adapted from Cobb, N.B., Fast optical and process proximity correction algorithms for integrated circuit manufacturing, Ph.D Dissertation, University of California, Berkeley, California, 1998 With permission.)

(a) (b)

FIGURE 35.9 (a) and (b) Line-end pullback and (c) and (d) corner rounding (Reproduced from Schellenberg,

F.M., Zhang, H., and Morris, J., Optical Microlithography XI, Proceedings of SPIE, 3334, 892, 1998 With

permission.)

pitch and therefore a narrow spectrum, which passes easily through the pupil, while isolated features with sharp edges correspond to a range of spatial frequencies, including many high frequencies that are cut off by the pupil It is therefore not a surprise that isolated and dense features of the same nominal dimension may have different images on the wafer

Additional effects that can impact the image are line-end pullback and 2D corner rounding, illustrated in Figure 35.9 [40] These also are interpretable partly through the spectral analysis of the layout

To compensate for the loss in higher spatial frequencies, the positions of the edges in the original layout can be altered and adjusted as appropriate to correct the image in the local environment [38,42,43] This is illustrated in Figure 35.10 Additional features not present in the original layout, sometimes called ‘scattering bars’ or ‘assist features’ can also be added to the layout [44,45] These features, with dimensions chosen so that they themselves do not print on the wafer, form a quasi-dense environment around printing features, which would otherwise be isolated An example is illustrated

in Figure 35.11 The overall effect is to make the behavior of the isolated features better match the behavior of dense features on the final wafer

35.2.3.2 PSM

Traditional photomasks are fabricated using a lithography process to etch away portions of a layer

of opaque chrome coated on a quartz mask blank [46,47] The presence or absence of chrome forms the pattern to be reproduced on the wafer However, the underlying quartz substrate of the mask can

be etched as well Because the refractive index of the quartz and air are different, a relative phase shift between the two neighboring regions can be created This is illustrated in Figure 35.12 For apertures that are close together, if the light emerging from the apertures has the same phase, the images overlap on the wafer, the fields add, and the spots blur together, as shown in Figure 35.13a

If the phase difference is 180◦, as shown in Figure 35.13b, however, the wave peaks and troughs sum

to zero in the overlap region, and destructive interference occurs Therefore, for two regions in close

(a) (c)

(d) (b)

FIGURE 35.10 (a) Original layout and (b) its simulated wafer result, and (c) layout after modification with

OPC and (d) its simulated wafer result The wafer result for the corrected version is clearly a better match

to the original drawn polygon (Adapted from Maurer, W and Schellenberg, F.M., Handbook of Photomask Manufacturing Technology, S Rizvi, Eds., CRC Press, Boca Raton, Florida, 2005 With permission.)

FIGURE 35.11 Example of a contemporary layout with printing features and SRAF (Reproduced, Courtesy

Mentor Graphics.)

proximity, a dark fringe forms, allowing the images to remain distinct [48–50] This is often called

an ‘alternating’ PSM, because the phase alternates between apertures

Careful assignment of the mask regions to be etched, or phase-shifted, can lead to enhanced resolution for an IC layout This can lead to problems, however, if polygons that require phase shifting

180 ⬚

0 ⬚

Quartz substrate

FIGURE 35.12 Cross-section view of an optical wave passing through two apertures of a photomask Etching

the mask substrate for one of the apertures can produce a phase shift of 180◦

0 ⬚ 180 ⬚

FIGURE 35.13 Amplitude and intensity for (a) a conventional mask, and (b) a mask with a 180◦phase shift Contrast for neighboring apertures is clearly enhanced for the phase-0 shifting mask (Adapted from Maurer,

W and Schellenberg, F.M., Handbook of Photomask Manufacturing Technology, S Rizvi, Ed., CRC Press,

Boca Raton, Florida, 2004 With permission.)

in one area of the chip are contiguous with polygons in other regions that require the opposite phase These topological constraints, illustrated in Figure 35.14, are called ‘phase conflicts’, and can place additional design rule restrictions on layouts [51–54]

Several variations on phase-shifting techniques have been adopted The most common is a hybrid phase shifter, called an ‘attenuated PSM’ [55] Here, the opaque chrome material of a conventional photomask is replaced with an attenuating but partially transmitting material (typically a MoSi film with 6 percent transmission [56]), with properties selected such that the light weakly transmitted through the film emerges with a phase shift of 180◦ This improves contrast between light and dark

regions, because the E-field (and therefore intensity) must be zero somewhere near the edge between

the clear region and the phase shifted, darker region However, fabrication techniques are similar to regular chrome mask processing, and no additional quartz etch step is required

There are also several double exposure techniques, in which certain phase-shifted features are created on a first photomask, while a second mask is used to trim or otherwise adapt the exposed

Desired layout

Phase conflict regions

180 ⬚ phase shift

FIGURE 35.14 Examples of layouts that have phase conflicts.

region to complete the exposure [57–60] In this way, some of the unwanted artifacts of the phase-shifting structures can be eliminated in the second exposure More details on various PSM techniques can be found in the literature [35,37]

35.2.3.3 OAI

For light falling at or near normal incidence to the photomask (on-axis illumination), the diffraction spectrum is straightforward to interpret For light entering at an angle (i.e., using off-axis illumination [OAI]), the spectrum is shifted [61], as shown in Figure 35.15 Clearly, depending on the layout on the mask and the imaging properties of the lens system, the spectral content of the image can be significantly affected

FIGURE 35.15 Spectrum for an off-axis ray (left) and spectrum for an annular cone of off-axis rays (right).

+1 orders

0 orders

− 1 orders

−

FIGURE 35.16 Spectrum for conventional illumination (left), and for off-axis annular illumination (right),

in which the annulus has been chosen to coincide with the diffracted orders of the pattern on the photomask

Typical illuminators shape the light to be uniform and to illuminate the photomask with a fairly narrow range of angles The spectrum of illumination then corresponds to a circle By using illumi-nation with a specific angle of incidence, represented, for example, by the annulus in Figure 35.16, certain pitches can be emphasized and their imaging contrast enhanced, but only at the expense of lower contrast for other spatial frequencies [61,62] For IC layouts with a large proportion of periodic patterns, such as memories, a suitable choice of illuminator pattern that matches the spatial frequen-cies of the layout can enhance imaging performance significantly [62] An example of this is shown

in Figure 35.17, in which a quadrupole-like illuminator was used in combination with subresolution assist features (SRAFs) to overcome certain “forbidden pitches” of low contrast [63]

More elaborate interactions between the spectrum of source angles and the photomask layout are possible Shown in Figure 35.18 is an example of an IC cell and a source spectrum created through mask/source co-optimization There are several methods demonstrated to achieve this goal [64–67]

35.2.3.4 RET Combinations

Although each of these techniques can enhance lithographic performance in and of itself, it is

in combinations that dramatic improvements in imaging performance are achieved For exam-ple, phase-mask images may have higher contrast, but still suffer from iso/dense bias, requiring OPC [59] Likewise, combinations of OAI tuned for photomasks with OPC layouts can be very effective [68–70] In some cases, all three techniques have been used together to create the best lithographic performance [35,71,72] Success with developing processes using these combinations, with choices tuned to the unique combinations of skills present in individual companies, is a lively source of competition among IC makers