aerial image formation

7/29/2019 Aerial Image Formation

1/20

Photolithography:

The Basics of Projection Printing


2/20

Photolithography

Photolithography is at the Center of the WaferFabrication Process

*

Thin Films

Implant

Diffusion Etch

Test/Sort

Polish

PhotoPatterned

wafer


3/20

Photolithography Types


4/20

Projection System

Chris A. Mack, Fundamental Principles ofOptical Lithography, (c) 2007

4

Light Source

Condenser Lens

Mask

Objective LensWafer

Block diagram of a generic projection imaging system


5/20

Projection System


5

Light Source

Condenser Lens

Mask

Objective LensWafer

Illumination SystemIllumination system delivers light to the maskSufficient intensity

Proper directionality

Proper spectral characteristics

Adequate uniformity across the field


6/20

Projection System


6

Light Source

Condenser Lens

Mask

Objective LensWafer

Illumination System

Mask

changes the transmittance of the light

Causes diffraction

Diffraction


7/20

Projection System


7

Light Source

Condenser Lens

Mask

Objective LensWafer

Illumination System

Objective lens

Picks up a portion of the diffraction pattern

Projects the image onto the photoresist

Image Formation


8/20

Diffraction Analysis


8

Light Source

Condenser Lens

Mask

Objective LensWafer

Illumination System

Mask

changes the transmittance of the light

Causes diffraction

Diffraction


9/20

Diffraction

Theory of propagation of light

Solution to Maxwells Equations: complicated

Diffraction Integrals for far field: simpler

Light propagation in homogenous medium

FresnelDiffraction

Region

(z w)

KirchhoffDiffraction Region

(z > /2)

FraunhoferDiffractionRegion

(z w2/)

Fraunhofer Diffraction Integrals


10/20

Fraunhofer Diffraction Integral

Mask Transmittance Function:

How does the mask transmit light?

Requires the solution of Maxwells equations for

that material: complicated!

Kirchhoff boundary condition

Feature size > 2 && chrome thickness < /2

Ignore diffraction caused by mask topography

),( yxtm

chrome0

regionnttranspare1),(

yxtm


11/20

Fraunhofer Diffraction Integral

maskat theincidentfieldElectric:),(

functionmittancemask trans:),(

mediumtheofindexrefractive:

planendiffractiomask tofromdistance:

lightofhwavelengt:

planendiffractio:plane''

planemask:plane

yxE

yxt

n

z

yx

yx

Given

m

dxdyeyxtyxEffTyfxfi

myxmyx )(2),(),(),(

z

nyf

z

nxf

ff

ffT

where

yx

yx

yxm

',

'

patternndiffractioofsfrequencieSpatial:,

patternndiffractiotheoffieldElectric:),(

Fourier Optics

The diffraction pattern is

just the fourier transform

of the mask pattern

transmittance!

)),(),((),( yxtyxEFffT myxm


12/20

Diffraction Examples


12

x00

mask

01

Tm(fx)

tm(x)

Two typical mask patterns, an isolated space and an array of equal lines and spaces, and

the resulting Fraunhofer diffraction patterns assuming normally incident plane wave

illumination. Both tm and Tm represent electric fields.

x

xxm

f

wffT

)sin()(

j

x

x

xxm

p

jf

f

wf

p

fT )()sin(1

)(


13/20

Chris A. Mack, Fundamental

Principles of Optical

Lithography, (c) 2007

13

0.0

0.2

0.4

0.6

0.8

1.0

-3 -2 -1 0 1 2 3

w fx

Intensity

Magnitude of the diffraction pattern squared (intensity) for a single space (thick solid

line), two spaces (thin solid line), and three spaces (dashed lines) of width w. For the

multiple-feature cases, the linewidth is also equal to w.


14/20

Image Formation: Imaging Lens


14

Light Source

Condenser Lens

Mask

Objective LensWafer

Illumination System

Objective lens

Picks up a portion of the diffraction pattern

Projects the image onto the photoresist

Image Formation


15/20

Imaging Lens

The diffraction pattern extends throughout

the x-y plane

The objective lens (finite size) can only capture

a part of this pattern

Higher spatial frequencies lost


16/20

16

Aperture

(x, y)-plane

ObjectPlane

max

ObjectiveLens

EntrancePupil

maxsinnNA

NA

p

n

z

nyf

n

z

nxf

y

y

xx

min

1

sin'

sin'

NA/ : The maximum spatial frequency that can enter the lens

Large NA => larger portion of diffraction pattern is captured => better image

NA

KR

1


17/20

Formation of the Image

Lens Pupil Function

Mathematical description of the amount of the

light entering the lens

Lens performs the inverse fourier transform of

the transmitted diffracted pattern

NA

ff

NAffffP

yx

yxyx

22

22

,0

,1),(

),(),(),(),(),(

1

yxyxM ffPffTFyxE

yxgyxgFF


18/20

18

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

-w -w/2 0 w/2 w

Horizontal Position

Rela

tiveIntensity N = 1

N = 3

N = 9

Aerial images for a pattern of equal lines and spaces of width was a function of the

number of diffraction orders captured by the objective lens (coherent illumination). N is

the maximum diffraction order number captured by the lens.


19/20

PSF: Point Spread Function

Means of characterizing the resolving

capability of an imaging system

Normalized image of an infinitely small

contact hole

NANAR

JPSF

rNA

JffPFyxE

ffTyxt

ideal

yx

yxmm

7.0to66.0

)2(

)2(),(),(

1),()0,0(),(

2

1

11

0.0

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

Normalized Radius = rNA/

0.2

0.4

0.6

0.8

1.0

NormalizedIn

tensity


20/20

Oblique Illumination

Plane wave strikes at an angle

Phase differs for different points on mask

Shift in the position of the diffraction pattern

NA

kR

ffPffffTFffyxE

f

eyxE

yxyyxxmyx

x

xi

2

)},()','({)',',,(

'sin'

),(

1

1

'sin2

Diffraction Pattern

Lens Aperture

Mask Pattern(equal lines and spaces)

aerial image formation

Documents