First basics of optics and diffraction

Question 1

Photolitography improvements

Answer 1

1. Reduction in wavelength of the UV light source.
2. Increase in numerical aperture.
3. Chemically amplified DUV resists
4. Resolution enhancement techniques (e.g., phase-shift masks and optical proximity correction)–>k1 factor reduction
5. Wafer planarization (chemical mechanical
   planarization, or CMP) to reduce surface topography.
6. Advances in photolithography equipment (e.g., stepper and step-and-scan).

Question 2

Exposure tool evolution

Answer 2

• Contact (1:1) = mask in contact with wafer (image ratio 1);
• Proximity (1:1)= mask next/close to wafer (image ratio 1)
• Projection (1:1) = projection lens between mask and wafer (image ratio 1);
• Projection (5:1) stepping = projection lens between mask and wafer (image ratio 5);
• Projection (4:1) step and scan = projection lens between mask and wafer (image ratio 4);
• Direct writing= no mask, direct writing in resist using eBeam or laserbeam

Question 3

Exposure tool evolution: contact vs proximity

Answer 3

CONTACT
Advantages:
-not complex (compared to projection);
-cheap system;
-fast: wafer exposed at once.
Disadvantages:
-the mask contacts the wafer so mask wear and contamination: need of periodic cleaning to avoid repeating defects on subsequent wafers;
-no magnification;
-mask usually the same size as the wafer: large and expensive

PROXIMITY
Advantages:
-fast: wafer exposed at once;
-the mask does not contact the wafer so no mask wear or contamination.
Disadavantages:
-mask separated from the wafer: greater diffraction leads to less resolution;
-mask usually the same same of the wafer: large and expensive.
Neither:
-more complex or more expensive

Question 4

Exposure tool evolution: 1x projection aligners

Answer 4

During the mid 1970’s optically based mask generation tools began to reach their limits because 1:1 masks needed to grow in size to match the larger wafers. Moreover the masks didn’t last very long because they were physically in contact with the wafer and so would get damaged and pick up defects fast.
Later on (1980–1990), projection systems were introduced, in which the image of a mask is projected by an optical system onto the wafer. The distance between mask and wafer became extremely large and contamination problems due to physical contact were no longer a concern.
Initially the masks remained 1x and contained the complete image to be projected onto the wafer, although only part of the image was exposed simultaneously to keep the complexity of the optics under control.
Mask and wafer are then moved simultaneously, so the synchronization between the wafer and mask scanning had to be very accurate.

Question 5

Motivations for optical projection litography

Answer 5

As soon as optical lithography started to be based on projection systems, using an optical “lens” between the mask and the wafer, the principle of today’s optical lithography has been established and still remain unchanged for many years.

Light diffration with lens:

• diffracted light collected by the lens (optical lens can collect diffracted light and enhance the image);
• less diffraction after focus by the lens (short wavelength waves have less diffraction).

Question 6

Optical projection basics

Answer 6

Let’s consider a generic projection system.
It consists of a light source, a condenser lens, the mask, the objective lens and finally the resisti-coated wafer.
The combination of light source and condenser lens is called the illumination system.
In optical design terms, a lens is a system of
(possibly many) lens elements. Each lens element is an individual piece of glass (refractive or dioptric element) or a mirror (reflective or catoptric element). The purpose of the illumination system is to deliver light to the mask (and eventually into the objective lens) with sufficient intensity, the proper directionality and spectral characteristics, and adequate uniformity across the field.
The light then passes through the clear areas of the mask and diffracts on its way to the objective lens.
The purpose of the objective lens is to pick up a portion of the diffraction pattern and project an image onto the wafer, which, one hopes, will resemble the mask pattern.
The first and most basic phenomenon occurring here is the diffraction of light.

Question 7

Diffraction

Answer 7

It is the fundamental phenomenon that is responsible for image formation in optical litography. The diffraction theory simply describes how light propagates and this propagation includes the effects of the surroundings(boundaries).

Question 8

Huygen’s principle

Answer 8

Any wavefront can be thought of as a collection of radiating point soures (spherical secondary wavelets). The new wavefront at some later time can be constructed by summing up the wavefronts from all of the radiated spherical waves (envelope of these wavelets).
Following Huygen’s principle, electromagnetic fileds can be thought of as sums of propagating spherical or plane waves.

Question 9

Fresnel diffraction theory

Answer 9

Joseph Fresnel: diffraction mathematical theory. A summation turned into an integral phase of light when adding the propagating spherical waves so he basically added the concept of interference discovered by Young into the original Huygens principle.

Question 10

Huygens-Fresnel principle

Answer 10

Huygens-Fresnel: every unobstructed point of a wavefront, at a given instant, serves as a source of spherical secondary wavelets with the same frequency as that of primary wave. Amplitude of optical field at any point beyond is the superposition of all these wavelets considering their amplitudes and relative phases.
The occurrence of diffraction depends on the relative size of aperture.
When the wavelength of incident light is in the order of aperture, the unobstructed point in the aperture acts as secondary source and the waves will spread out at large angles into the region beyond obstruction. If the incident wavelength is much smaller than the aperture, diffraction effect can only be observed within a short distance behind the aperture and beyond this region, shadow begins.

Question 11

Kirchhoff principle of diffraction

Answer 11

Kirchhoff: he put in a more rigorous footing Huygens’ scalar diffraction theory. Fresnel’s formulas are in fact a simplification of Kirchhoff’s formulation for the case of distance away from the diffracting plane (i.e. the distance from the mask to the objective lens) much greater than the wavelength of light.

Question 12

Fraunhofer principle of diffraction

Answer 12

Fraunhofer: if the distance to the objective lens is very large.

Question 13

Coherently illuminated mask

Answer 13

With a coherently illuminated mask, meaning the illumination is only from one single direction, the image in intensity is the aerial image.
Tipically with coherent illumination, fringes are created in the diffuse shadowing between light and dark, a result of interference. Only when there is separation between the obstacle and the recording plane rectilinear propagation occurs: CONTACT EXPOSURE.
As the recording plane is moved away from the obstacle, there is a region where the geometrical shadow is still discernible. At close distances, where geomtric shadowing is still recognizable, near-field diffraction, or Fresnel diffraction, dominates: PROXIMITY EXPOSURE.
Beyond this region, far from the obstacle, the intensity pattern at the recording plane no longer resembles the geomtrical shadow, but contains areas of light and dark fringes. In the far-field Fraunhofer diffraction dominate: PROJECTION EXPOSURE.

Question 14

What is contact exposure?

Answer 14

When talking about a coherently illuminated mask, fringes are created in the diffuse shadowing between light and dark, a result of interference.
Only when there is separation between the obstacle and the recording plane rectilinear propagation occurs: CONTACT EXPOSURE.

Question 15

What is proximity exposure?

Answer 15

As the recording plane is moved away from the obstacle, there is a region where the geometrical shadow is still discernible. At close distances, where geomtric shadowing is still recognizable, near-field diffraction, or Fresnel diffraction, dominates: PROXIMITY EXPOSURE.

Question 16

What is projection exposure?

Answer 16

Beyond the region where the geometrical shadow is still discernible, far from the obstacle, the intensity pattern at the recording plane no longer resembles the geomtrical shadow, but contains areas of light and dark fringes. In the far-field Fraunhofer diffraction dominate: PROJECTION EXPOSURE.

Question 17

Near field/Fresnel diffraction for contact/proximity exposure

Answer 17

Assuming near field, so lambda> lambd.

• Strong influence of the gap g

Question 18

Far field/Fraunhofer diffraction for projection exposure

Answer 18

Assuming far field: Wmin^2<

Question 19

Airy disk

Answer 19

When talking about far field/Fraunhofer diffraction and the image of an ideal point source diffracted through a circular aperture at a long distance g. The bright central maximum is known as Airy’s disk. The bright central disk is surrounded by a number of fainter rings. Neither the disk nor the rings have intensities that are defined sharply but instead are shaded at the edges. The rings are separated by circles of zero intensity. About 85% of the energyentering the optical system is concentrated in the Airy disk, while the other 15% is spread through the rings (Airy’s pattern).
The Airy disk is only valid for a large distance g, where Fraunhofer diffraction applies, but calculation of the shadow in the near-field must rather be handled using Fresnel diffraction. However the exact Airy pattern does appear at a finite distance if a lens is placed after the aperture then the Airy pattern will be perfectly focused at the distance given by the lens’ focal length (assuming collimated light incident on the aperture).

d=diameter of the focusing optics
f= focal length of focusing optics
n= index of refraction (normally 1 in air but now 1 is used in immersion litography)
alfa= angle to the edge of focusing optics
From geometry: d=2[nfsin(alfa)]

Question 20

Rayleigh criterion

Answer 20

It is for barely resolving two objects (i.e. edges of a pattern feature) that are point sources of light. The center of the Airy disk for the first object occurs at the first minimum of the Airy disk of the second. This means that the angular resolution of a diffraction-limited system is given by:
R=1.22flambda/d=1.22flambda/n(2fsinalfa)=0.61lambda/nsinalfa=0.061lambda/NA

NA is the numerical aperture of the focusing optics and it describes the focusing strength of the projection system. However all our derivation was based on an ideal “point source” so we can generalie using a constant K1:
CD=k1*lambd/NA

Question 21

Numerical aperture

Answer 21

It is one of the key input parameters in optical litography togeter with the exposure wavelength lamda and the coherence factor.
In optical litography it is defined as the sine of the maximum outcoupling angle of light exiting a lens, measured at the wafer, multiplied by the refractive index of the material where the image is formed.
NA=n*sin (alfa)
For air the refractive index is 1, but for water immersion litography it increases to 1.45 for 193nm litography.
Furthermore the numerical aperture is the ability of a lens to collect diffracted light so a lens with larger NA can capture higher order of diffracted light and generate sharper images. Basically we can see NA as a band pass filter cutting high diffraction angles.

Question 22

Far field/Fraunhofer diffration: general case

Answer 22

Let’s describe the electric field transmittance of a mask pattern as m(x,y), where the mask is in the x,y plane and m(x,y) has in generala both magnitude and phase. For a simple chrome-glass mask, the mask pattern becomes binary: m(x,y) is 1 under the glass and 0 under the chrome.
x’y’ plane is the diffraction plane, i.e. the entrance to the objective lens, and z is the distance from the mask to the objective lens. We assume monochromatic light of wavelength lambda and that the entire system is in air (so n=1 can be dropped).
The electric field of our diffraction pattern E(x’,y’) is given by the Fraunhofer diffraction integral:
E(x’,y’)=doppio integrale tra -infinito e +infinito di m(x,y)exp(-2pii(f_x+f_yy)) dxdy
where f_x=x’/zlamda and f_y=y’/zlambda are called the spatial frequencies of the diffraction pattern.
It is simply a Fourier transform!!! Thus the diffraction pattern (i.e. the electric field distribution as it enters the objective lens) is just the Fourier transform of the mask pattern.
If we take two examples of mask patterns, one an isolated space, the other a series of equal lines and spaces, both infinitely long in the y-direction then the resulting mask pattern functions, m(x,y) look like a square pulse and a square wave.
The isolated space gives rise to a sinc function diffraction pattern, while the equal lines and spaces yield discrete diffraction orders.

Question 23

Diffraction pattern for equal lines and spaces

Answer 23

For a mask pattern with a series of equal lines and spaces, the resulting mask pattern function m(x) is a square wave.
The graphs of the diffraction patterns use spatial frequency as the x-axis and since z and lambda are fixed for a given stepper, the spatial frequency is simply a scaled x’ coordinate.
At the center of the objective less entrance (f_x=0), the diffraction pattern ha s abright spot called the zero order which is the light that passes through the mask and is not diffracted. The zero order can be thought of as a sort of “DC” light, providing power bt no information regarding the sie of the features on the mask (basically it is the average value of the mask pattern).
To either sides of the zero order are two peaks called the first diffraction orders and these peaks occur at spatial frequencies on n/p, where p is the pitch of the mask-

Question 24

Diffraction pattern for equal lines and spaces

Answer 24

For a mask pattern with a series of equal lines and spaces, the resulting mask pattern function m(x) is a square wave.
The graphs of the diffraction patterns use spatial frequency as the x-axis and since z and lambda are fixed for a given stepper, the spatial frequency is simply a scaled x’ coordinate.
At the center of the objective less entrance (f_x=0), the diffraction pattern ha s abright spot called the zero order which is the light that passes through the mask and is not diffracted. The zero order can be thought of as a sort of “DC” light, providing power bt no information regarding the sie of the features on the mask (basically it is the average value of the mask pattern).
To either sides of the zero order are two peaks called the first diffraction orders and these peaks occur at spatial frequencies on n/p, where p is the pitch of the mask-

Question 25

Given a mask the electric field as it enters the objective lens is?

Answer 25

Given a mask in the x-y plane described by its electric field transmission m(x,y), the electric field M as it enters the obective lens is given by:
M(f_x,f_y)= F{m(x,y)}
where F represents the Fourier transform, f_x and f_y are the spatial frequencies and are simply scaled coordinates in the x’,y’ plane.

Question 26

Let’s follow the light as it enters the objective lens

Answer 26

In general the diffraction pattern extends throughout the x’-y’ plane but the objective lens, being only of finite size, cannot collect all the light in the diffraction pattern: only those portions of the mask diffraction pattern that fall inside the aperture of the objective lens go on to form the image.
An useful way to describe the size of the lens aperture is considering the maximum angle of diffracted light that can enter the lens.
Light passing through the mask is diffracted at various angles. Given a lens of a certain size placed at a certain distance from the mask, there is some maximum angle of diffraction alfa, for which diffracted light just makes it to the lens. Light emerging from the mask at larger angles misses the lens and is not used in forming the image.
The most conveniente way to describe the size of the lens aperture is by its numerical aperture NA.
A large numerical aperture means that a larger portion of the diffraction pattern is captured by the objective lens, and for a small NA much more of the diffracted light is lost.

Question 27

How does the lens affect the light entering it?

Answer 27

We want to image to resemble the mask pattern.
Since diffraction gives the Fourier transform of the mask, if the lens could give the inverse Fourier transform of the diffraction pattern, the resulting image would resemble the mask pattern. This is the job of spherical lenses. But because of the finite size of the NA, only a portion of the diffraction pattern enters the lens: even an ideal lens cannot prodice a perfect image unless the less is infinitely big!
The case of an ideal lens, where the image is limited only by the diffracted light that does not make it through the lens, we call such ideal system “diffraction limited”.

Question 28

The pupil function

Answer 28

The pupil function of an ideal lens simply describes what portion of light enters the lens: it is 1 inside the aperture and 0 outside.
Thus, the produt of the pupil function and the diffraction pattern describes the light entering the objective lens.
A final expression for the electric field at the image plane (at the wafer) is:
E(x,y)=F^-1{M(f_X,f_y)P(f_x,f_y)}
The aerial image is defined as the intensity distribution at the wafer and is simply the square of the magnitude of the electric field.

Basically the pupil function is used to mathematically describe the behaviour of how the NA describes the maximum angle of diffracted light that enters the lens.

Question 29

Full imaging process

Answer 29

First, light passing through the mask is diffracted and the diffraction pattern can be described as the Fourier transform of the mask pattern.
Since the objective lens is of finite size, only a portion of the diffraction pattern actually enters the lens. The NA describes the maximum angle of diffracted light that enters the lens, and the pupil function is used to mathematically describe this behaviour.
Finaly, the effect of the lens is to take the inverse Fourier transform of the light entering the lens to give an image that resembles the mask pattern. If
the lens is ideal, the quality of the resulting image is only limited by how much of the diffraction pattern is collected. This type of imaging system is called diffraction limited

Question 30

Far field/Fraunhofer diffraction: partially coherent illumination

Answer 30

So far we have assumed that the mask is illuminated by spatially coherent light meaning that the light striking the mask arrives only from one direction. We have also assumed that the coherent illumination on the mask is normally incident and the result was a diffraction pattern centered in the entrance to the objective lens.
However, actual litography tools use a more complicated illumination where light strikes the mask from a range of angles (called partially coherent).
If the light strikes the mask at some angle teta’ the effect is to shift the diffraction pattern with respect to the lens aperture (in terms of spatial frequency the amount shifted is sin teta’/lambda).
This shift in position of the diffraction pattern can have a profound effect on the resulting image.
If f’_x and f’_y are the shift in spatial frequency due to the tilted illumination, the electric field at the image plane, i.e. at the wafer will be:
E(x,y,f’_x,f’_y)=F^-1{M(f_x-f’_x,f_y-f’_y)P(f_x,f_y)}.

Question 31

Partially coherent illumination: range of angles

Answer 31

A range of angles will cause a range of shifts, resulting in broadened diffraction orders: higher amount of energy collected at the lens aperture so better pattern imaging!!
The most common way to characterize the range of angles used for the illumination is the partial coherence factor, defined as the sine of the half-angle of the illumination cone divided by the objective lens NA, or the condenser lens NA divides by the projection lens NA, so the source diameter divided by the lend diameter.
Thus it is a measure of the angular range of the illumination relative to the angular acceptance of the lens, expressing the relative capture range of the condenser lens with respect to the projection lens, both seen at the wafer plane.

Question 32

Incoherent vs coherent vs partially coherent illumination

Answer 32

Incoherent: if the range of angles striking the mask extends from -90° to 90° (that is, all possible angles), theillumination is said to be (σ = infinite).
Coherent: When σ is zero
Partially coherent illumination: finite nonzero values of σ.

Most projection systems used in litography have coherence factors in the range 0.4-0.8.

Question 33

Image calculation of the partially coherent

Answer 33

It is based on dividing the full source into individual point sources. Each point source is coherent and results in an aerial image given by:
E(x,y,f’_x,f’_y)=F^-1{M(f_x-f’_x,f_y-f’_y)P(f_x,f_y)}.
The full aerial image is determined by calculating the coherent aerial image from each point on the source and then integrating the intensity over the source.