Optical Imaging in Projection Microlithography

6.2 Vector imaging 11' According to the scalar Fresnel-Kirchhoff diffraction theory (see §3.4), the field at a point Po caused by a spherical wavefront on a surface S converging toward P0 is: U(Po) =

ƒƒ

+ikR

A (Ps) e R dS,

(6.8)

s

where A (Ps) is the amplitude distribution on S, and R is the distance between the spherical surface and Po. When the semi-aperture angle of the spherical wave at the image point is large, the vector nature of these image-forming rays must be considered. The field should be represented by a vector: E(Po)

=

ƒƒA

e +ikR (PS) R dS.

(6.9)

s

The vector field E(Po) is a function of the vector amplitude distribution A(Ps) on S. The extension from scalar diffraction [Eq. (6.8)] to vector diffraction [Eq. (6.9)] is straightforward because the Fresnel-Kirchhoff diffraction formulae apply to all components of the electric field vector. Denoting the electric field vector by its Cartesian components E(Po) = [EE(Po),Ey(Po),EE(Po)],

the intensity is 1 (Po) = nagelE( Po) I 2 = nimage [Ex (Po) +E(Po) +Ez (Po)],

where nimage is the refractive index of the image space. iiiThe exposition in this section follows Yeung [42] and Flagello [43].

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Vector imaging



103

TM

N' Ps TM } T'

M'

TE

T' A

T/ i3Po N

i

TE T l E0=(EEO) M

nimage

R

S ^

nobject

Z

Figure 6.2: The field at Po is the vector sum of contributions from point sources on

S. To find the intensity I(Po) we need to determine the electric field components E, Ey , and EZ [42-44]. Consider the situation shown in Fig. 6.2, where a light ray AB, initially traveling in the z-direction in a medium with refractive index nobject, becomes the image-forming ray PSP0, with PS lying on the spherical wavefront converging toward the image point Po, situated in an ambiance with refractive index nimage . Let us define the meridional plane as the plane that contains the optic axis (the z-direction) and the ray PSPO. We can decompose the electric field vector into two components, one with its direction of vibration perpendicular to the meridional plane (orthogonal to the plane of the figure) and one that vibrates in the meridional plane (in the plane of the figure). We call the component that vibrates perpendicular to the meridional plane the transverse electric (TE) or the perpendicular component, since the electric vector vibration is transverse to the meridional plane. The other component we call the transverse magnetic (TM) or the parallel component» The TM direction of the image-forming ray PSP0 (NN' in Fig. 6.2) depends on the angle 19 between the ray and the optic axis. This direction is generally different from the TM direction of the original ray (MM' in Fig. 6.2). But the TE direction (TT' in Fig. 6.2) is independent of any rotation of the light ray by the imaging system. To find the contribution of the ray PPP to the vector field at Po, we need to determine how the electric field vector on the original light ray AB maps onto the imaging ray PSPo. Let us represent the propagation direction of the ray PSP0 by its direction cosines (a, 13, y). Illustrated in Fig. 6.3, these parameters are given by a=sin1coscp,

(3=sin19sincp,

and

y= cos t9.

The corresponding wave vector is k = k (a, ß, 7) . 1 °The transverse electric polarization is also called the s-polarization, and the transverse magnetic polarization is sometimes called the p-polarization.

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104



Oblique Rays

The perpendicular direction s" 1 (the TT' direction in Fig. 6.2) can be expressed in terms of the direction cosines. Since s" 1 is perpendicular to the meridional plane, we can define _ kxz S1

Ikxil

'

giving S1x = l ß Sly = —

1—YZ

,

S1 =0.

Denoting the fraction of x-polarized light that aligns with the perpendicular direction by Px 1, we have Px1 = x s1= In a similar manner,

_ _

_

l ß Y2

a

Py1 — y s1 1 — f

If E0 = nimage /nobject (Ex,Ey) are the Cartesian components of the electric field

vector of the original ray AB, the amplitude of the TE component of the ray PSPP is (assuming the imaging system does not modify the amplitudes of light rays) a

ß

E1=EP±+E. -L=Ex 1—I —Ey

1— .

Let us denote the contribution of this perpendicular component to the field at Po by E(Po;E1). We can decompose this quantity into the three Cartesian components: E1 ( 1 )1

rEPxi.x+EyPy

E(Po;E1) = E1(S1 . Y) = Ex Pxly+EyPy1y E1 (S1 . Z ) EXPX_, + EyPyiz where 13 2 PL', = Px1(S1•X) = Px1y

a Pj'

1—y2 y '

= &L (S1 ' y) = — 1

Px1z = Px1(S1 • Z) = 0,

= P1(S1'X) ^ ß =— 2 1-7 y

aß _ Y2 , PyIy = Py1(S1 ' y)



a2 (6.10)

Py_l. r = Py1(S1 • Z) = 0.

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Vector imaging



105

z

y

x Figure 6.3: The direction cosines of a ray vector represents its propagation direc-

tion. Each parameter PP11 can be interpreted as the fraction of i -polarized light that is aligned with the TE direction, and which is further decomposed into the j Cartesian component at the image point. For the TM component, we can define the parallel direction s by sly - k x sl, resulting in ay III X = 1_y2

S IIY

=

(6.11)

ßy

sjJ z Following a development similar to the TE polarization we obtain (see Exercise 6.8) a2 Pxllx = l_a

aßy

PYIIx = 1a ß

72

PxIIY_ 1 72'

ß2 Y PYIIY__ 1 _ 72'

Pxllz =

PYIIZ = - ß•

(6.12)

Summing the TE and TM polarizations, the contribution of an image-forming ray [traveling with direction cosines (a, 13,y)] to the vector field at the image point

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Oblique Rays

106

is proportional to PxJ+Pxllx E(Po) = Pxj y +Px II Y

r

PXilz

omMo3, 0MoYY

oxZ MoYZ

PY-L,+PYllx ^

PYIY +PYIIY PYIIZ

Exy]

(6.13)

r

;YI = MoEo.

Since the direction cosines are related to the normalized frequency variables by a = f sin9 obj,

=gsineob^, y = 1— (/2 + g2 ) sine A obi = 1— pé, we can consider the matrix Mo to be a function of the normalized frequency variables: namely, Mo (a, 3,y) = Mo (.f , g) In terms of Mo (/, g), the vector amplitude distribution A (Ps ) in Eq. (6.9) is A(Ps) =A(Ps)Mo(f,g)Eo•

A(Ps) is the amplitude distribution on the spherical wavefront S (see Fig. 6.2) and Mo(/, g) Eo represents the vibrational direction of the electric field vector. The intensity can be expressed as

j(x,9) = nimage f ... fJ(/(/+/' +' )H * (.f +.f",g+g') 0 (/',g')Mo(i+j',ó+g')Eo

•O *

(u ' g")M (/+.f",8+8")E i

*

*

e—tz^[(f _ f„)z+(g _g )Y] d fdgd/'dg d/" d g = nimage 1T.

f jj,g)H(f +f',g+g )H * (f +f",g+g')

i ={x,y}

Moik(/+I', +ó')Mo jk (.f +.f",g+g")EiE^

j={x,y}

k={x,y,z}

„ 6 (/',g )O*(.f"^g')e—ti2^[(f,_ f )x+(g—S )y]d f'dgd/"dg'. (6.14)

The extension from scalar partially coherent imaging to vector imaging involves replacing the scalar function ô(/, g) in Eq. (4.35) by the vector entity 6(f, g)Mo(/,g)Eo• Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms

Wave propagation across a dielectric interface

107

photoresist

wafer stack

Figure 6.4: A resist image is formed by coupling of electromagnetic energy into the photoresist layer.

In the scalar imaging scenario, two waves Ó(f', g') and Ó* (f", g") would interfere to form a modulated image if TCC(f', g'; f", g") is nonzero. For vector imaging, however, not only must the effective source f (f , g) and the displaced pupils H (f + f', g + g') and H* (f + f", g +g") overlap, the vectors Ó(f', g')Mo (f + f', g +g')E0 and 6* (f", g ")M* (f + f", g + g")E* should not be perpendicular. Image quality deteriorates with increasing misalignment between these two vectors. We shall explore this phenomenon of polarization-induced image quality degradation in more detail in §6.7.

6.3 Wave propagation across a dielectric interface Equation (6.14) applies to the situation where the imaging rays propagate in a homogeneous medium. We term such an image the coupling image. The coupling image is called the aerial image if the image space has a refractive index sufficiently close to 1, and it is called the aquatic image if the medium is water. In optical projection lithography, the apposite image is that formed in the photoresist, which is an organic, light-absorbing layer within a stack of materials coated on a silicon wafer. Energy is deposited into the photoresist by electromagnetic wave coupling through the wafer stack, as illustrated in Fig. 6.4. We call the intensity distribution within the photoresist the resist image. In transmission through the dielectric stack, the amplitudes and propagation angles of the light rays vary. To derive the intensity distribution within the photoresist, we need to account for these changes by considering wave propagation through the wafer stack. Although topography on a wafer is in general irregular, we restrict our analysis to stratified media in which the material properties vary only in the z-direction. This approximation is usually adequate since most layers in a modem integrated circuit process are planarized. In this section we study wave reflection and transmission across a dielectric interface. The results will be generalized to

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108



Oblique Rays

zinc

rL ninc aL

erefl

all

z_0

Srefl

rll erefl

X

n einc

ntran

etran

Y

t1

til Stran

z

-- X Z

Figure 6.5: Reflection and refraction of a plane wave.

stratified media in §6.4.

6.3.1 The laws of reflection and refraction When a plane wave incidents upon a boundary between two homogeneous materials, as illustrated in Fig. 6.5, two secondary waves are produced: a reflected wave that propagates back into the incident medium and a wave that is transmitted into the second medium (see Exercise 6.10). If we denote the propagation direction of the incident wave by s", the incident wave varies according to r

sint

(6.15)

—t

cinc

where ci nc is the speed of propagation in the incident medium. The reflected and transmitted waves can be represented in a similar manner by the vectors s efl and s"t• Let the material interface be parallel to the xy-plane (the surface normal n aligns with the z-direction) with the boundary located at z = 0. The incident, reflected, and refracted waves must have the same variation on this boundary plane:

r

r , Sinc

Srefl

r , Stran onz=0,

—t

zinc

Cinc

(6.16)

Ctran

with taan being the speed of propagation in the second medium. Since Eq. (6.16) applies to all values of x and y, it implies int Sx

=

Cinc

reff Sx

Iran

= SX and

zinc

Ctran

int Sy zinc

reff

=

Sy

tran

= y S

zinc

( 6.17)

Ctran

Let us define the plane formed by si nt and the surface normal n to be the plane of incidence. It follows from Eq. (6.17) that both srefl and stro lie on this plane. "

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Wave propagation across a dielectric interface

109

To determine the relationship between the different waves, assume that the xzplane is the plane of incidence. Denoting the angles that and stra" make with the z-axis by 0 inc erefl, and 6 tran , we have Sinc =

rs n c1 r sine1 inc S inc = 0

Srefl =

Y Sz c [cos ejnc

and

Stran =

Sz fl

S refl

=

SZ fl

1

Y

rs'n _ [sinOtran stran S trap j [cos 0

I

I

sin erefl COS Orefl

(6.18)

where, according to the convention in Fig. 6.5, s'z ` = cos Sine > 0, SZ fl = COS Orefl C 0,

(6.19)

and Szran _ cos, >0. Substituting Eq. (6.18) into Eq. (6.17) gives sin Oi nc = sin 9refl _ sin Ott„ (6.20) Ctran cinc Cinc

The first equality in Eq. (6.20) and the first two relations in Eq. (6.19) imply that Orefl = lt — eint or Orefl = eint,

(6.21)

where 0refl + Orefl = it (see Fig. 6.5). This equation and the result that s efl lies on the plane of incidence constitute the law of reflection. The second equality in Eq. (6.20) means that

sin eint — cinc — ntran sinetran Ctran ninc '

(6.22)

where nip, is the refractive index of the incident medium and n traz, is that of the second material. The relation in Eq. (6.22), together with our earlier conclusion that stra" lies on the plane of incidence, express the law of refraction. It is also known as Snell's law.'

6.3.2 Reflected and transmitted wave amplitudes Let us derive expressions for the amplitudes of the reflected and transmitted waves assuming that the two media are nonconductive. For each of the incident, reflected, `'In optically anisotropic materials, each incident ray in general gives rise to two refracted beams. This phenomenon of birefringence is discussed in Appendix A.

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110

Oblique Rays

and transmitted waves, we can resolve the field into components parallel and perpendicular to the plane of incidence, as illustrated in Fig. 6.5. Let a be the amplitude of the incident electric field; this quantity is in general complex, with the phase being the constant term 0 in Eq. (1.33). The Cartesian components of the electric field of the incident wave are Einc

—all COS ei nc e il3inc

I

E fc = Emc = [

Y

Eint

aiei^mc

all sinei,cei,5in^

Z

[

where embeds the spatial and temporal dependence (k • r — cot) of the incident field, and the subscripts and I denote the parallel and perpendicular components of the field, respectively. Since, according to Eq. (1.26), Sin' X E inc _ TincHinc

the magnetic field components are cos 0m c en^

c —al

Hinc = Hync

Hine

I

— 1

—ale

Tlinc

sm eince "5inc

z a1

Similar expressions can be derived for the transmitted and reflected waves: —rij cos

Erefl =

—r1 cos 0refle ii4^fl

ereflel^ree Hrefl =

r1ei^4refl

r1 sin 0refle '

—rlleil4ree

1

Tinc

j

rl sin 0refle z

—til cos 6 „e` l5ff^

E tran =

—t1 cos 0ei 1 u^

1

Htran _

t1eii9ua„

—tlleißu^a

Iltran

tl sin 8 tran e`^tran

t1 sin 6 tran e` t

where r and t are the reflected and refracted wave amplitudes, respectively. The boundary conditions as a consequence of Maxwell's Equations (see Exercise 6.9) dictate that the tangential components of E and H to be continuous across the material interface. Therefore E nc + Erefl —_ Etran

Eint + Erefl —_ Etran Y Y Y

Hint + Hxrefl

Hinc

—

Htran

Y

Hrefl _— Htran Y

(6.23)

Y

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111

Wave propagation across a dielectric interface

On substitution of the expressions for the field components, Eq. (6.23) becomes (all — rll) cos 8; ßc = tll COS Orra,,, a1+r1 =t1,

1 (a1— rl) cos Ainc = 1 Tlinc

t1 cos 6tran,

(6.24)

1ltran

Tlinc 1(all + r11) rltran III The expressions in Eq. (6.24) fall into two groups; the first and fourth equations contain only components parallel to the plane of incidence, whereas the other two relate components perpendicular to the plane. The parallel and perpendicular waves are independent of each other. They can be studied separately. Solving Eq. (6.24) gives the reflection and transmission coefficients of the parallel and perpendicular components: _ rl _ line cos einc — TÍtran cos 8 , P II — all lÍineCOSeinc+TtranCOSetran'

z

ll — all P1 =

t1

2 ltran COS O inc line cos eine + 1ltranCOSetran '

rl

11tran cos eint — Tlinc COS etran

a1

iitran cos eint +Hine COS Otran ' 21jtran COS e ine

'L1=—=

a

(6.25)

1 11tr an

cos eine + 11inc cos Otran

These Fresnel's formulae can take a different form when the media under consideration are nonmagnetic (u r = 1), as refractive indexes can be used in place of characteristic impedances: _ ntran cos eine — nine cos e tran P 11 ntran COS eine + ni e COS etran' 2ninc COS eine till= n traz, cos eine + ni,, c cos Otran '

nine _ cos e ine — ntran cos e

(6.25')

P1 nincCOSBinc+ntranCoSetran 2n c cos e ine

'C1

nine cos eine + ntran cos etran

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112



Oblique Rays

Using the law of refraction [Eq. (6.22)], Eq. (6.25') can be expressed in terms of the angles ei n and e (see Exercise 6.11): tan(e tran — einc)

PM = tan(e tran +e ;nc )'

2 sin etran cos eine 'III sin (etran + e ine) COS(etran — eine) P1

'

(6.25")

e

sin(etran - ine) sin (etran +eine

)'

2 sin 8tr an cos eine sin(etran + eine)

For normal incidence, eine = 0 and hence e tra„ = 0; Fresnel's formulae become _ _ nine — ntran Pi— nine + ntran '

2n inc nine + ntran

Pi=

(6.26)

nine — ntran

nine + ntran 2n

til

= nine +incntran

There is no distinction between the two polarizations. Although Fresnel's formulae have been derived under the assumption that the plane of incidence is the xz-plane, the equations are valid for an arbitrary plane of incidence, provided the electric field vector is decomposed into components perpendicular and parallel to the plane of incidence. In general, the electric field vector in the perpendicular direction has nonzero x- and y-values, whereas the parallel direction has nonzero values in all three Cartesian components (see §6.2).

6.3.3 Reflectivity and transmissivity According to Eq. (1.28) the light intensity in nonmagnetic materials is I = SJ = nT1ole^ 2 . The energy flux incident onto a unit area of the dielectric interface is I

ine = ISincIC OSeinc = nincTÍola1 2 coseinc)

and the reflected and transmitted energy fluxes leaving the boundary are Irefl = ninc

1 o r 2 COS eine and

Itran = ntranijo It l 2 COS 0tran.

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Wave propagation across a dielectric interface



113

Let us examine how the incident energy is divided between the reflected and transmitted waves. We are interested in the fraction of energy reflected and transmitted:

i r12

jrefl

!d

an

R=—=— and T== lal2 jinc

niran cos e tran It I2 Hint COSeinc

lal t

,

(6.27)

where R is the reflectivity and T is the transmissivity. In accordance with energy conservation, their sum is 1 (see Exercise 6.14): R+T= 1. As the amplitudes of the reflected and transmitted waves depend on the incident polarization, let us decompose the incident electric field vector E 1Re into parallel and perpendicular components all

=

a1 = a sin aine,

cos ai ne and

a

where lElneI = ^al. The vibration angle of the incident field a;n, is the angle between the plane of incidence. Let us define the quantities as

Eine and

jnRe = nincllO lall I 2 cos Dine = jine COS 2 ainc, jlc = nincTl0la1 1 2 cos eint j1Í fl

= jinc sin t

ainc,

= nine 110 I rij I 2 co5 eine,

jlfl = nincrl0I rl I 2 COS eine

With these notations,

jreg R jinc

=

jrefl + jrJefl

jrefl

jine

jrefl c COS 2 ainc + jnc

II

sin ainc (6.28)

1

= RII cos t ainc + R l sine ainc, where jrefl e 2

2

= ta (etran — eint) ( = I P I I tan 2 (0tran + eine)

= I1 II R

6 . 29 ) 6.29

and I1fl 2 R1

sin2(Otran —eint)

= jinc = I Pl I = sin 2

(6tran +O)

( 6 .30 . 30 )

Analogous quantities for the transmitted wave are similarly derived: Jtran T = jinn = TII cos t aine + T L sin t ainc, Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms

Oblique Rays

114

with TII

I = sin(26tra„) sin(29jnc) _ ntra„ cos Otr ar, 2 (6.31) ninc COS 0 inc II Sint (Oran + eine) COS2 (egan — eine)

T1 _ ntran COS 6 tran It± 22 _ 1 nine COS eine

sin(28 tmn ) sin(29 (6.32) sint (etran + eine) )

Since I p± I I p l l I and I tiL 10 I r l l I in general, the reflected and transmitted light can be partially polarized even if the incident light is unpolarized. The degree of polarization (see §6.1) is Prefl = R1— R II R1 + RII ptran = T1—T11

T11 At normal incidence, there is no distinction between the parallel and perpendicular polarizations. In this scenario R = RII = R _ uinc — ntran 1— \ nine + ntran

2

(

and

T = T TL = 4ntran (nine + ntran) 2

Unpolarized light remains unpolarized upon reflection and transmission.

6.3.4 Polarization upon reflection and transmission The vibration angle a changes upon reflection and refraction. Denoting the vibration angles of the incident, reflected, and transmitted electric field vectors by ainc, arefl, and atr., respectively, a 1 r1

tan abc = — ,

tan arefl = — ,

tan at rm =

-

Using Fresnel's formulae [Eq. (6.25")], we obtain tan arefl = —

COS(A tran — eine)

tan a; n ^,

(6.33)

COS(Otran + eine)

e

tan atran = COS (O tran — ine) tan ainc .

(6.34)

Because 0 < 9i,, c and 6tran < (n/2), I arefl I> I ainc

I

and

I atran I C I ainc I

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115

Wave propagation across a dielectric interface

_inc

rl ninc

a-L

r11=0

Z_O all

ntran

Srefl

`

erefl

n

X

0inc `

etran

Y

tL t11

tran

z

X

z

Figure 6.6: For incidence at Brewster's angle, the reflected light has no component

in the plane of incidence. The reflected and transmitted beams travel in perpendicular directions.

The plane of vibration turns away from the plane of incidence upon reflection, whereas the electric field vector turns toward it on refraction. The denominator of Eq. (6.33) is infinite when 6 uaz, + Bine = Tc/2; in this scenario, the reflected electric field vector is normal to the plane of incidence. R^1 = 0 as a result. This situation occurs when (see Exercise 6.15) an tan GBrewster = nam

mc

(6.35)

nine

The angle BBrewster is the Brewster angle (also called the polarizing angle). If light is incident at this angle, the electric vector of the reflected light has no component in the plane of incidence. The degree of polarization [see Eq. (6.7)] of the reflected wave is P = 1, independent of the degree of polarization of the incident light. In this situation, the reflected and transmitted waves are perpendicular to each other, as illustrated in Fig. 6.6. 6.3.5 Total internal reflection When the incident medium is optically denser than the refracted medium (nine > such that, according to Snell's law, n), there is an incident angle n eincernaf

etran

=

îc /2

= arcsin tran

and

nine

For a wave incident at Oi nc = eincerna^ the refracted wave travels parallel to the dielectric interface. There is no energy flow from the incident medium into the refracted medium. At incident angles larger than Oin tema , the quantity sin 6 uß„ = nine sin Oi nc (6.36) ntra„

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116

Oblique Rays

is greater than 1. The angle 6,, becomes complex and cos 9 t,a„ is purely imaginary: cos 9 traz, = 1— sin e e

=

i

Hint Z ( n ) sin 2 9inc —1. trap

(6.37)

Assuming the plane of incidence coincides with the xz-plane, the phase factor of a refracted harmonic wave is e+i(kxx+kz z—c,)t) = e — iwt e +ik(xsinO tran +z COS O n ) nz

= e — i(At e +ikX( fliflC sLn9ince

—kz H an s1I1z9inc -1

Sometimes called evanescent waves, the beam propagates parallel to the interface (the x-direction under our assumption); its amplitude decreases exponentially in the surface normal direction (the z-direction under our assumption). The reflection coefficients in situations where ' mc > e nternal can be deduced from Eq. (6.25") (see Exercise 6.13): P11 =

cos Sine — in n 2 sin 2 00i —1

cos O + in n2 sin 2

()

—1

(6.25")

n COS eint — i,/ 2 sin2 e int — 1

P1 = 2 n cos eint+l n 2 sm

ein,. -1

with n = ninc /nt an . The magnitudes of both reflection coefficients are 1:

TAAI =IP I = 1. All incident light is reflected. We call this phenomenon total internal reflection. Although fields exist in the refracted medium, there is no net energy flow across the interface. This lack of energy flow is confirmed by the time-averaged normal component of the Poynting vector just across the surface (see Exercise 6.19): <S•n>=0.

(6.38)

6.4 Stratified media`' 1 The properties of stratified medium are constant throughout each plane perpendicular to a fixed direction. Taking this fixed direction to be the z-direction, the material properties has only a z-dependence, namely, £e = E e (Z)

and

,um = ,um (z)

v i lnterested readers may wish to consult Berning [45] for more details on the theory and calculations of optical thin films.

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Stratified media

117

Consider a plane, time-harmonic electromagnetic wave propagating through this stratified medium. Because any such plane wave can be decomposed into two independent components (see §6.3.2), one having its electric field vector vibrating in a direction perpendicular to the plane of incidence, and the other having its magnetic field vector perpendicular to the plane of incidence, we will examine them separately. Similar to our investigation of polarization in §6.2, we call the former situation the transverse electric (TE) polarization (the perpendicular polarization), and the latter the transverse magnetic (TM) polarization (the parallel polarization). The symbols I and denote whether the electric field vector is perpendicular or parallel to the plane of incidence.

6.4.1 Basic equations As in §6.3, let us assume, without loss of generality, that the xz-plane is the plane of incidence. In the TE polarization, the electric field is nonzero only in the ydirection. For a plane, time-harmonic electromagnetic wave, Maxwell's equations [Eqs. (1.1)—(1.4)] become

_o

aHz

ay

aHy —

JHx

aHz

az

ax

az

aH, _ —=0,

aHy — aHx

ax

(6.39)

= —iCOEe Ey ,

aHx ^y = 0 ,

ay — 0

-á a y = io umHx,

(6.40)

O= iWW,u m Hy

Hy =0,

= icou Hz .

ax

(6.41)

These equations show that the electromagnetic field has no y-dependence. Differentiating Eq. (6.40) with respect to z and differentiating Eq. (6.41) with respect to x result in z dLm = t o)

Hx + 1O Um

(6.42)

áx

and 2

áE

y

a

= kau m a z .

(6.43)

Substituting Eqs. (6.42) and (6.43) into Eq. (6.39) gives the following partial differential equation:

2E + aEy +n2k°Ey = d(ldz ) adzy.



(6.44)

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118

Oblique Rays

With a proposed solution of the form Ey (x,z;t) =X(x)U(z)e ` wt -

Eq. (6.44) becomes

1

a 2 X(x) — 1 a 2 U(z) d(lnµm ) 1 aU(z) 2+n k2

' X (x)

ax 2 2 U(z) az

az

dz U(z)

o

(6.45)

Since the left-hand side of Eq. (6.45) depends only on x, while the right-hand side depends only on z, they must both be equal to a constant. Denoting this constant by S2 ko, we obtain`' i'

X

(x) = cx e"

0 x,

with cX being a constant,

and E, (x, z; t) = U(z)e` ( Skox

-wt)

For a plane wave traveling at an angle 0 with respect to the z-axis in a medium of refractive index n, (6.46) c, = n sin 6. To solve for U (z), we define the functions V (z) and W (z) such that Hx (x, z; t) _ — 1

1 dU(z) ,( *X -wt) = V (z) e i(skox-wt) e /Am um dz

from Eq. (6.40),

and Hz (x,z;t) _

5o U(z)ei(s/cax -wt)

=

W( z ) e i(0x -0)t)

from Eq. (6.41).

Using these two expressions and Eq. (6.39), we can deduce the following crosscoupled equations between U (z) and V (z):

dU(z) _ — dz

icoµm V(z)

(6.47)

and

=i

(;2k 2/Am —>

ddzz)

e)

U(z).

(6.48)

Upon differentiation with respect to z and cross-substitution, these equations become d2U(z) _ d (lnic m dU(z) 2 2 o (S — n Z) U (z) + k d z dz dz2 )

(6.49)

viiTe general solution is a superposition of two waves: X(x) = cx,e`y`0x +cg2 e —i ch0x

Under the current consideration of one plane wave, only the first term is retained.

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Stratified media



119

and Zk2

d2 v ( z ) — d[ln { z (; 1m —£e)] dv(z) -

dz

dz 2 dz

+ß(S2 —n 2 )v(z).

(6.50)

Since both U (z) and V (z) satisfy a second order differential equation, they can be expressed as linear combinations of two particular solutions, namely, U (z) = c 1 Ui (z) + c u2 U2 (z) and

V (z) = c 1 VI (z) + cV 2 V2 (z) . These particular solutions [Ui (z), Vi (z)] and [U2 (z), V2 (z)] are coupled by Eqs. (6.47) and (6.48), meaning Vl (z) dU2 (z) — dUl (z)

dz

dz

V2 (z) = 0

and Ul (z) dV2 (z) — dVl (z)

dz

dz

U2 (z) =0.

Together they imply C^Z

[Ul(z)v2(z)_U2(z)V1(z)] =0

(6.51)

or

Ul (z)V2 (z) — U2 (z) V, (z) = constant.

6.4.2 Characteristic matrix To simplify the mathematics, let us pick particular solutions that satisfy the following conditions: U1(0) = V2(0) = 1

and

U2(0)=V1(0)=0.

(6.52)

With these conditions, the solutions of Eqs. (6.49) and (6.50) with U(0) = Uo and V(0) = Vo can be written as U(z) = Ul (Z) Uo + U2 (Z)Vo

and V(z) =V1(z)Uo+V2(z)Vo•

We can express these equations in one matrix equation: Ul (z) U2(z)

Uo = NQ(0). vo [v (z) I — [vi(z) (z) V2

Q(z) = U(z)

(6.53)

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120



Oblique Rays

The determinant of N is 1 (see Exercise 6.20). Inverting the above matrix equation

gives Q( 0 ) =N —' Q(z) _

1 Vi (z) Ui(z) —

)

1

(6.54)

IV (z)1 MQ(z)

The determinant of the matrix M is also 1. Given M, knowledge of the fields at a particular z-plane is sufficient for determining those at other planes. The matrix M is the characteristic matrix of the stratified medium. Let us derive the characteristic matrix of a homogeneous dielectric film, with a TE plane wave propagating in a direction making an angle 0 with the z-axis. In this situation, S = n sin 0 [see Eq. (6.46)], and Eqs. (6.49) and (6.50) become d'-U(z) 2 2 dz2 +kón cos 0U(z) = 0, d'-V(z) a d`2

+ kón cosa0 V (z) = 0.

The solutions are U(z) = cl cos(konzcos0) + c2 sin(konzcos0), V(z) = i

m

cos0[ —cl sin(konzcos0)+c2cos(konzcos0)].

The particular solutions that satisfy the conditions stated in Eq. (6.52) are

Ui(z) = cos(konzcos0), U2(z) = 1 ^m sin(konzcos0), icosøV E e Vi (z) = —i cos 0 £e sin(konz cos 0), V2(z) = cos(konzcos 0), and the characteristic matrix is

M (z)

-1

Ui (z) V(Z) Vi

-

)1

cos(konzcos6)

pose £e sin(konzcos6)

i cos 6m sin(konzcos 0)

cos(konzcos A)

(6.55)

j

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Stratified media



121

Since Maxwell's equations are invariant under the simultaneous substitutions .ii E H

and

Ee

-

/Lm,

the characteristic matrix for a TM wave is cos(konzcos0) MTM(z) _

—icos8 £e sin(konzcos0)

^os`e µm sin(konzcos0) cos(konzcos 0)

with Hy(x,z,t) = U(z)e` (5kox-0)r) and Ex (x z; t) = V

(z)ei(Skox—wt)

The sign change of the off-diagonal elements originate from replacing —icqu m by iWEe in Eq. (6.47). We can therefore define a general characteristic matrix as M(z) —

cos (konz cos0) [ixVsin(kOnzcosO)

, sin(konzcos8)1 cos(konz cos 0) JI '

(6.56)

with NJ = cos 8 £

e

= cos 8

in the TE polarization, and

/1m

(6.57)

yr = —cos 8 ^m = —rl cos 8 EQ

in the TM polarization.

Since the tangential components of the electric and magnetic vectors are continuous across dielectric interfaces (see Exercise 6.9), the characteristic matrix of a homogeneous dielectric film can be extended to describe a stratified medium comprising two materials, the first extending from z = 0 to z = zl and the second from z = zi to z = z2• If M1(z) and M2(z) are the characteristic matrices of the two materials, then and

Q(0) =Mi(zl)Q(zi)

Q(zi) =M2(z2 — zi)Q(z2),

and Q(0) =Mi(zi)M2(z2 — zi)Q(Z2).

This expression can be generalized to a stratified medium consisting of q layers with a total thickness of d, with di denoting the thickness of the ith layer: Q(0) = MI(di)M2( d2)...Me(dq)Q(d) nMi (di ) [

Q(d)

i=

(6.58)

= M(d)Q(d)

= `m 11 m 121 rU(d)^

LM21 m22

lV (d)

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122



Oblique Rays

Wave transmission through the entire stratified medium can be characterized by the four matrix entries ml1, m12, m21, and m22.

6.4.3 Reflection and transmission Let us now examine the reflection and transmission of an electromagnetic wave upon incidence onto a stratified medium, as illustrated in Fig. 6.7. Consider first the TE polarization, in which the electric field vector is polarized in the y-direction. Following the approach taken in §6.3.2, the electric field in the incident medium is E =E + E refl = aIe i13inc + rle il^refl Y y y

The variables 1j and i3refl embed the spatial and temporal dependence of the incident and reflected waves. It follows that U(0) = a1 + rL. Using Eq. (1.26), we can write V(0) = 1 Vinc ( — a1 + r1),

where, as defined in Eq. (6.57), cos einc Winc =

Tlinc

The transmitted fields are U(d)=t1

and

V(d)=—yr.tL.

Equation (6.58) becomes [Vinc( a_L+r_L)l r — [m21 m22] ^—Want] . —

Solving this equation results in r1 _ Vine (m11 — 1IJtranml2) + (m21 — Wtranm22) (6.59) Nlinc(m11 — 19tranml2) — (m21 — Wtranm22) ' P1 = a1

_

t1

_

21Vinc

(6.60)

T1 a1 zinc (m11 — Wtranml2) — (m21 — Wtranm22)

In the situation where the stratified medium is absent (see Fig. 6.5),

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Stratified media



123

inc

refI

ninc

aL

Xr,

11

ereil ereil

,

a II

d

n P =n'ß,(1+ivP ) a P 0 r layerp 0 0

bt

0

n q =n'Q (1+

I

Z=O dp

k su sac

I dq J

ginc

ntran

etran

Yr X

tL

1

Z Z

til gtran

Figure 6.7: Reflection from and transmission through a stratified medium.

and —

Pi

Tltran COS e inc — 1linc COS 9tran 11tran COS ejnc + Tlinc COS etran

21jtran COS e inc 11tran COS Ainc + 1 1inc COS Atran

We recover Eq. (6.25). The matrix equation for a TM wave can be similarly derived: (a ll + rll)

cos 6inc(all — rll)

mll m12

1_"^tII

m21 m22

cos Ot tll

Solving this equation gives rij 1Vinc(ml1 — tranm12)+(m21 —ltranm22)

P11=— all

(6.61)

1 Vinc(MI1 — Nftranm12) — (m21 —1 Vtranm22)

tij —2tltran cos 6inc ( ill= — = tinc(mil — Wtranml2) — (m21 — Wtranm22)

(6.62)

all

where yr is defined in Eq. (6.57). Although Eqs. (6.59)—(6.62) have been derived under the assumption that the plane of incidence is the xz-plane, they are applicable for arbitrary planes of incidence, provided the electric field vector is decomposed into perpendicular and parallel components.

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124

Oblique Rays

6.5 Intensity distribution in photoresist`'nl Referring to Fig. 6.7, suppose that the photoresist, with a thickness dp, is the p th layer of a stratified medium consisting of q layers. To determine the resist image, let us define a substack that is a stratified medium comprising the layers (p + 1) to q of the original stratified medium, and an angle 6 p between the z-axis and the +z-propagating plane wave in the photoresist corresponding to an incident angle 6;ßc such that ninc sin eint = nimage Sin eine = np sin Op,

np = , (1 + iKp ) .

The amplitude of the +z-propagating TE wave in the photoresist a1 is related to the incident TE amplitude al by a P ,substack = a 1 I

stack 1 I

where the superscript "substack" refers to quantities related to the substack and the superscript "stack" denotes quantities associated with the entire stratified medium. The electric field within the photoresist layer arising from an incident TE plane wave of amplitude a 1 is stack

ikpfl•r _ ti1 ikP`•r E1— al substack [e + p1 substacke

I

1

stack

= ei(kPx +kP) a ti l ikp1—z [ e+(d1) +

1

substack

1

_ kP,

substack e — ikp(d,—z) p1 ]'

where kP e (kp, kP) and kp fl = ( kP, k, —kf) are the wave vectors of the +z traveling and —z-traveling plane waves in the photoresist. Since the parameters tack , tïbstack, , ,? stack , and kp are functions of B ob] and z only, the ratio of the photoresist electric field amplitude to the incident amplitude is stack

ELI

ti l ikz ( p — z) + p s bstacke— ikP(dp e+' a1 ,substack =[ 1

—z)1 = IF±(Pe, z) I.

(6.63)

1

Equations for the TM polarization can be similarly derived. According to the convention of Fig. 6.7, the x- and y-components of the reflected wave subtract from the incident wave, whereas the z-components are summed: E'Y a

ll

I=I =I

,stack

substack [ e

+ikp(dp—z) — p 5tack e — ikP(dP

—z)1 = IF (pe,z)I

(6.64)

IFI (p e z)I.

(6.65)

I

EZ ,stack

[e+d_z) +p llubstack e — ikP(dd —z)1 =

substack a ll Ts J v iii The

exposition in this section follows Yeung [42] and Flagello [43].

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Intensity distribution in photoresist



125

The effect of the stratified medium is a scaling of the amplitude of each imageforming ray of the spherical wavefront by the factors FL, Fr', and F, depending on the incident angle of the light ray and the z-position within the photoresist. The quantities PP11 and FIIj in Eq. (6.13) should be scaled by the appropriate factors. For example,

P X —^ F1PX j,

and

PYllz

Fj PYllz

The vector field at an image point P arising from a ray is a modification of Eq. (6.13):

Pxi Py F1 Fxy 0 0 01 ' x 'yMX E(P) = 0 0 Fi FxY 0 Px1y PYLY

0



0

Msxx Ms = Mss, Ms

0

0

FI

PxIIY PYIIY PXllz PYIIZ

^Ey J (6.66)

1 Ex = MstackEo,

M&Z Msy^

where Ms.(po;z) = FLPX +F Px l x Msyx(pe;z) = F1Py ix+F PYIIx Ms, (pe; z) = FLPXLy + FXYPxIIY , Msyy (00; z) = FLPy 1y + F PYIIY, (6.67)

Msxz (Po; z) = F^ Pp,

Msy, (Pe; z) = F, PYIIz.

The resist image intensity is

I(,9)=nP f. f iu,g) 7 (ƒ+f^,g+g)H * (f+f^^,g+g^) O(J ,g )Mstack(f +.f' , '

'

+S' ;z)Eo .6* (1" , ")M Ck(f + J", + ";z)E e -^ 2n[(f'-f")x+(g'- ')y] d fdgd f, d g d j" d g

=nP ...ƒju,g)H(f+.P,g+g)H*(f+f^^,g+g^) i ={x,y} j={x,y}

Msik(f + f^ g+g^;z)Ms;k (f + f" g+g";z)EiE^

k={x,y,z}

6 (.jß , S )O * (.Í n

g') e -

^2 ^ [( f_ f)X+(s^- )yl d fdgd J' d g d f"dg'.

(6.68)

Because of reflections from the photoresist interfaces (characterized by F L , FIx, and F^ ), the resist image exhibits a standing wave whose ratio decreases with resist

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126

Oblique Rays

Figure 6.8: (a) Reflections from material interfaces create a standing wave in the resist image. (b) Post-exposure baking reduces the variation, resulting in a smoothed latent image. depth due to absorption (KP ). ïx An example resulting from the exposure conditions detailed in §D.4.1 is shown in Fig. 6.8(a). Post-exposure baking smooths any standing wave that may be present such that the latent image corresponding to the resist image of Fig. 6.8(a) may resemble that shown in Fig. 6.8(b).

6.6 Immersion imaging A reason for using light rays at large angles with respect to the optic axis is resolution, since both minimum half-pitch and minimum dimension decrease with increasing numerical aperture NA = n; n,age sin 8ob^ . In addition to using a large semi-aperture angle 9 obj, we can increase the numerical aperture by immersing the image space in a high-refractive-index medium [47].x Figure 6.9 shows a schematic representation of an immersion imaging system. This diagram is the same as Fig. 2.7, except that the image space refractive index is nage > 1 and that the focal length of L2 is increased by a factor of nimage• Following the analysis in §2.5 and noting that the angles (po and (pi in Fig. 6.9 are related by Snell's law [see Eq. (6.22)] sin (p0 = ni. ge sin pi, (

i "The

standing wave ratio can be reduced by decreasing reflection from both photoresist interfaces (using top antireflective coating and bottom antireflective coating) as well as increasing resist absorption [46]. The latter approach, however, degrades resist image quality. 'For example, water with a refractive index of 1.45 is a suitable medium at 193 nm.

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Imaging with oblique rays



127

Figure 6.9: Representation of an immersion imaging system.

the lateral magnification is

yi



Mlateral = y

_ nimage fM sin (p i

f

sin (po

M'

the angular magnification Mang„lar is

sini5i _ y /nimagefM 1 an a ular — sin150 nimageM' y'/f '

g

and the longitudinal magnification is Mlongitudinal = Mlateral/Mangular = nimageM

2

Compared with an imaging system with the same numerical aperture in which the object and image media are the same [see Eq. (2.12)], light rays in an immersion system travel at shallower angles. The obliquity factor is modified from Eq. (4.11) accordingly: l^g^M2 pé = (6.69) E(0) = 4

1—pé

6.7 Imaging with oblique rays In the scenario shown in Fig. 6.10, where two light rays interfere to form an image, the intensity distribution depends not only on the angle between the rays [see Eq. (5.20)] but also on their polarization. With transverse electric polarization, the vibrational directions of both beams are the same. The total field is the scalar sum of all constituting fields. In the transverse magnetic polarization, however, the field

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128



Oblique Rays

bi

a1

bll

all

Figure 6.10: The fields add in a scalar manner in the perpendicular polarization but

add as vector quantities in the parallel polarization. vectors do not align. The fields arising from all beams add in a vector manner to give the overall field distribution. Since

images resulting from the parallel polarization have lower contrast than those resulting from the perpendicular polarization. 7 Let us consider, as examples that are shown in Fig. 6.11, the aerial images of three equal line-space chromium-on-glass patterns [see Eq. (5.7)] of different periodicities. The illumination for each pattern is an optimized dipole described by Eq. (5.17): f(j,g) = S(f+ 1 / 2PX)S(S)•

°

With px = 1 [Fig. 6.11(a)], the TE image shows a higher contrast than the TM image. The unpolarized image is the average of the TE and TM images, a result consistent with Eq. (6.5). When the period decreases to px = 1/v [Fig. 6.11(b)], the two interfering orders [see Fig. 5.7(a)] are mutually perpendicular. The parallel field components do not interfere; the TM image has no modulation. But the TE image still has a healthy contrast. With further reduction of the period to px = 0.6, the TM image exhibits intensity reversal [Fig. 6.11(c)]. The nominally bright area becomes dark and the opaque region becomes bright. The unpolarized image contrast is merely 0.277 despite a TE image contrast of 0.906. Although the fraction of p-polarized light increases as a beam travels through a wafer stack (see §6.3.4 and Exercise 6.17), resist-image quality degradation is less severe than the aerial image contrast loss depicted in Fig. 6.11. As a consequence of Snell's law (see §6.3.1), light rays in photoresists, with a typical refractive index of 1.8, travel at shallower angles, thus alleviating TM image degradation. Figure 6.12 demonstrates the difference between aerial and resist images for the scenario of Fig. 6.11(b), the situation where the TM aerial image has no modulation. Contrast of the TM resist image is 0.747 [Fig. 6.12(b)]. Such degradation of transverse magnetic image quality is sometimes said to be caused by polarization-induced stray light.

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Imaging with oblique rays

129

TE

TE TM

TM

0

T

cN

unpolarized

unpolarized^ 0 C

-0.50

-0.25

0.25

0.00

position (2/NA)

(a)



i3

=

0.50



-0.50

1

0.00

-0.25



0.25

0.50

position (A/NA)



(b)Px= 1 /\ TE TM

T

c

unpolarized

a, C

-0.50

0.00

-0.25

0.25

0.50

position (7/NA)

(C)

0.6

Figure 6.11: Transverse electric images show higher contrast than transverse magnetic images.

A TM resist image that has zero contrast or that exhibits intensity reversal is possible only if (see Exercise 6.24)

NA>

n^

(6.70)

Assuming a photoresist refractive index of 1.8, this condition means that NA> 1.273, a value that is not uncommon with immersion systems. In increasing the numerical aperture for resolution improvement, immersion imaging is more susceptible (than reducing the wavelength) to vector effects as light rays can travel at more oblique angles within the photoresist. In our analysis of image formation and adoption of the canonical coordinates [Eq. (4.30) in §4.5], we showed that optical imaging scales with XO/NA. But the 1 /NA dependence is first-order, accurate only if the fields add in a scalar manner. With oblique rays, vector addition of interfering light beams gives rise to a second-order phenomenon that increases in significance with vibration direction misalignment (namely, with NA) and that partly diminishes the first-order resolution improvement. The overall resolution improvement is sublinear with 1 /NA.

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130



Oblique Rays

9 TE

TM ..

TE

ó

TM _T

/

N

C N

\

/

/

\

0

-0.50

-0.25

0.00

0.25

0.50

-0.50

-0.25

0.00

0.25

0.50

position (A/NA)

position (AJNA)

(b) resist image

(a) aerial image

Figure 6.12: Resist image quality degradation is less severe than aerial image contrast loss. NA=0.1

R

NA=0.5

y, o

NA=0.9

N C
w

C •—

o

ó

O

-1.00

-0.50

0.00

0.50

1.00

position (AJNA) Figure 6.13: Unpolarized images of a space produced by three optical systems having the same XO/NA ratio. Figure 6.13 shows images of a space, described by Eq. (5.5) with px = 2, cl = 0.5, tfg = 1, and tbg = 0, produced by three optical systems. The ?/NA ratios of these systems are the same, but their wavelengths, and hence their numerical aper -tures,dif.Thymwtelosnuricapte(NA=0.1)roducs the sharpest image with the highest peak intensity. Image quality degrades with increasing numerical aperture. The image of the NA = 0.5 system is worse than that of the NA = 0.1 system, but it is better than that of the NA = 0.9 system. Given a choice of optical systems with the same 2O/NA ratio, the one with the lowest numerical aperture is preferred from the image quality viewpoint.

Exercises 6.1 Show that Eq. (6.1) describes a field vector whose end-point traces out an ellipse.

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131

Exercises

6.2 Show that Iµ < 1. 6.3 Show that 1(t9, 4)) in Eq. (6.4) is constant if, and only if, J, = Jyy and

'Cxy

= 0.

6.4 Show that the determinant of the coherency matrix is non-negative. 6.5 Determine the coherency matrix of circularly polarized light. 6.6 Equation (6.5) shows that unpolarized light is equivalent to two independent linearly polarized waves. Show that unpolarized light can also be considered as the sum of two independent circularly polarized light. 6.7 Derive the expressions of I 1po1, Ipolx> and Ip ,,ly in Eq. (6.6). 6.8 Derive Eq. (6.12). 6.9 Relate the normal and tangential components of an electromagnetic field across a dielectric discontinuity in the absence of currents and charges. 6.10 Show that the boundary conditions derived in Exercise 6.9 imply the existence of a transmitted and a reflected wave. 6.11 Derive Eq. (6.25") from Eq. (6.25'). 6.12 Referring to Eq. (6.25"), comment on the relative phase between the incident, reflected, and transmitted waves for nonmagnetic materials in which nin, and ntraz, are real numbers. 6.13 Deduce Eq. (6.25") from Eq. (6.25"). 6.14 Show that energy is conserved upon reflection and refraction by adding R and T in Eq. (6.27). 6.15 Using Snell's law, derive the Brewster angle [Eq. (6.35)]. 6.16 For a ray traveling in air (flair = 1), what is the polarizing angle for incidence upon glass (nglass = 1.5) and photoresist (np = 1.8)? What are the corresponding numerical aperture values? 6.17 A beam of unpolarized light is incident upon a flat air-glass interface at 30 deg. Determine the fraction of reflected energy, and the degree of polarization of the reflected light. 6.18 A linearly polarized light beam with a; 11, = 45 deg is incident upon a flat air-glass interface at 30deg. Determine the fraction of reflected energy. Compare this value with the result of Exercise 6.17. 6.19 Show that the time-averaged Poynting vector across a dielectric interface is 0 when total internal reflection occurs [Eq. (6.38)]. 6.20 Show that det(N) = det(M) = 1, where N and M are defined in Eqs. (6.53) and (6.54). 6.21 A wave traveling in a medium of refractive index ni ne is normally incident upon a stratified medium comprising only one layer with refractive index ns and thickness d. Derive the conditions under which the reflection coefficient is 0. Assume the refractive index of the material on the second side of the stratified medium is n tr 1.

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132

Oblique

Rays

6.22 Consider a stratified medium consisting of two layers, each of thickness ?/4. Their refractive indexes are (1 +A) and (1 — 0), where A « 1. Determine its reflectivity under normal incidence if nin, = n tran = 1. What is the reflectivity if n such pairs are stacked to form a stratified medium? 6.23 Given two imaging systems with the same wavelength, numerical aperture, and magnification, but having different coupling media of refractive indexes n( 1 ) and n( 2) Compare the obliquity factors of the two systems. 6.24 Derive Eq. (6.70).

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•

Chapter 7

Aberrations We have, thus far, assumed the image-forming wavefronts to be spherical. As we understand from Chapter 2, the emerging wavefronts from the pupil are in general aspherical, even if the lens surfaces are spherical. Deviation of wavefronts from sphericity is called aberration.' Within the domain of geometrical optics, an aberrated image can be examined by considering the intersection of aberrated rays with the image plane. Such treatment, adequate for instruments in which the aberrations are on the order of wavelengths, is known as the Seidel theory of aberrations. But the precision required of imaging systems in optical lithography means that the wave aberration is a small fraction of the wavelength. In this situation, the geometrical theory loses its validity; the Seidel treatment is insufficient. We need to study the effects of aberrations based on diffraction theory.

7.1 Diffraction of an aberrated wavefront Consider the imaging scenario shown in Fig. 7.1, where a spherical wavefront W of uniform amplitude and of radius R converges toward its center Po, which is defined as the origin. Let us denote the aberrated wavefront that intersects W at the center of the pupil by W', the points at which an imaging ray intersects W and W' by Q and Q', and the distance between Q and Q' measured along the ray by According to the Huygens-Fresnel principle [Eq. (3.11)], the field at a point P in the neighborhood of Po is U(P)

=

ff

e`k(R —rp) e +i2rz
dS.

(7.1)

S

`In addition to Born and Wolf [6], Mahajan [48] provides a detailed discussion on aberrations.

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134



Aberrations

rp



P (x,y,z)



Pr (x r ,y r )

Q

C^ Z

i

w'

po f' w

,

Po (x o,yo

eobi R.

Figure 7.1: Diffraction of an aberrated wavefront. Let (11, ) be the coordinates of Q, let (x, y, z) be those of P, and denote the radius of the pupil by po. We also define the variables (P, 0) and (r, yr) such that

1

x=rcoslV,

y=rsmyl,

1= POPcos0,

g =popsin0.

Since Q lies on the spherical wavefront, y=— R 2 —(PoP) 2 =—R[1—

(

pR2

2

+...1.

If R» po and R» x2 + y2 + z2 , we can make the approximation (see Exer-

cise 7.1) xll +y^+zX —kR—r n)__27t x R (

1 1

( 2R22 ] +p R rcos( ^ —tV)} P2 J

=-2n[prcos(c—y) —z (sin21Oobi where

x _ x _ x k/ sin 8 obj ko /NA' r = Á,/sin 8obj = x2 + y2 '

2)]'

k/ sin 9 obj ko/NA'

y

=

z k/ sint 6 obj

(7.2)

.

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(7.3)

135

General properties of the aberration function

Equation (7.1) becomes U(P)=Cl

ff fe -i2n ff e

i2 ic[prcos($ — W)+P2 f/2— z/sin 29 obi

1 o(P>$

,9)] dS

s 2n 1 = C2

[pYCOS(O-W)o,^ ,y?]

Pdf d^

00

The aberration function (Do (p, 0;1, y) generally depends on the coordinates of the image point. If we restrict our attention to an isoplanatic region (see footnote on p. 53), we can remove the dependence on land y and write (Do(p, 0;,x,y) = c(0, 0). The intensity Í'(P) is the square of the field." In the absence of aberrations, the principal maximum is located at the geometrical image point Po, with an intensity

of Let us normalize the intensity with respect to Jo:

jÛ(P)1 2 JO

_ 1 _

2t 1

1

Il

2

(7.4)

0 0

2n l 2

2

f e —i2n [prcos($ — W)+p 2z/ 2-0 (P,0)] Pdido

ff e +i2

dd ^

2 ,

00

with

2Z p . 15 = ID(P, 0) — prcos(0 — W) — 2

(7.5)

In the presence of aberrations, the location of the intensity maximum P is in general different from the geometrical image point Po. We call the point Pmax the diffraction focus. In addition to the intensity at the diffraction focus, we are often interested in the maximum normalized intensity on a particular plane, for example, at z = 0. This value is called the Strehl ratio.

7.2 General properties of the aberration function 7.2.1 Displacement theorem If c1 and C' are two aberration functions related by 0' _ O+Ap 2 +Bpcoso+Cpsino+D, 'We neglect the factor of 1/2 according to the convention described on p. 55.

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136



Aberrations

the variable i5 in Eq. (7.5) is = 0 — pr cos (4 — yr) — p 2 z/2 =«'—Ap 2 —Bpcos4—Cpsino—D—prcos(o—yr)—p 2 z/2 p g —D _ 0'—pr cos(O—]V) — 2

where 'cosy/ = P cosiƒ + B,

Y sinV' = Ysinyr+C,

(7.6)

The intensity distributions resulting from 0 and 0' are identical except for a displacement according to Eq. (7.6). 7.2.2 Intensity and average wavefront deformation From Eq. (7.4) the normalized intensity at the geometrical image point is 2n 1

I(Po ) = 2

ff 1

2

e +i2 "D(P>O) pd pd^

00

2n 1

=

2

I00 J f [1 + i27r0 -^-

(i2 it

2

+ ...1 Pd pd^ .

If the aberration is so small that we can neglect terms beyond the second power in (2nO), 2n 1

1(Po) = ßy2 J J (1 + i2nO —

2 2 47 )

0o

Denote by

Pd pdo .

(7.7)

J

the average of the n th power of c:

ff O" PdP d ^ f f pd pd4 Eq. (7.7) becomes

1(Po) = l + i27c- — 27r2 ^ 2 2 1 + 47r 2 (iS) 2 — 4n 2 ^2 = 1-4n2 [ I2 _( i5 ) 2 ] = 1— (2ic0cI, )2.

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(7.8)



Zemike polynomials

137

The quantity (O(D 11S ) 2 is the mean-square deformation of the wavefront:

ff(D— )20dpd0 _ ^ 2 — (j§) 2 . ff pd pdd

(D(D,ms) 2 =

(7.9)

For small aberrations, the intensity at the geometrical image point is reduced, compared with the nonaberrated intensity, by a fraction proportional to the wavefront mean-square deformation, independent of the characteristic of the aberrations. If we require that aberration should not cause the intensity at the geometrical image point to decrease by more than 5%, the root-mean-square deviation of the wavefront from sphericity should be less than

4(D1 < 0.0356. 7.3 Zernike polynomials To analyze the effects of aberrations on imaging, it is appropriate to express the aberration function 1 in terms of a complete set of polynomials, such that any reasonably well-behaved wavefront deviation from sphericity can be expressed as a series of polynomials of the set. Since we are concerned with circular apertures, the polynomial set should be orthogonal over the unit circle. It is also desirable that the form of the polynomials be unchanged with respect to rotation about the origin. The Zernike polynomials is a set that possesses these properties [49, 50]. Each polynomial of the Zernike set is a product of three terms: ZÍ (p ^) = an Rn (p)Y; ), (7.10) where

n_ [ -3+ 1+8j 1

I

2

m = n— 2 a

,^

L

(n+2)(n+l)-2j1 4

n+l (n + 1) (n m)/2

,

(7.12)

ifm=0, otherwise, (

—

R)=

(7.11)

'

-

1)S(n

—

s)I

pn-2s

s![(n+m)/2—s]![(n—m)/2—s]!

1

Y^ (0) = cos(m4)

sin(m4)





ifm=0, if m 0 and j is even, if m L 0 and j is odd.

The delimiters H•H represent the smallest integer that is greater than or equal to the delimited quantity; and the delimiters Li denote the largest integer that is less than Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms

138



Aberrations

Table 7.1: The first fifteen polynomials of the Zernike set.

172(4))

j

n m

1 2

0 1

0 1

1 p

1 cos 4)

piston x-tilt

3

1

1

p

sin4)

y-tilt

4

2

0

\

2p2 —1

1

defocus

x

Rn (P)

an

5

2

2

p2

sin 20

45° astigmatism

6

2

2

p2

cos 20

90° astigmatism

7

3

1

303 —215

sin4)

(balanced) y-coma

cos 4)

(balanced) x-coma

sin 34)

shamrock

cos30 1 cos20 sin 20 cos 44) sin44)

shamrock (balanced) spherical

8

3

1

f

3153 —215

9

3

3

10 11 12 13 14 15

3 4 4 4 4 4

3 0 2 2 4 4

v f

p; pl

f 10 10 10 10

6p`ß _6152+ 1 4p4 — 3p2 4p4 _3152 p4 p4

clover clover

or equal to the delimited quantity. The first fifteen Zernike polynomials are listed in Table 7.1. Additional terms are tabulated and plotted in Appendix C. The Zernike set described by Eq. (7.10) is orthonormal over the unit circle:

If

27[ 1

I

Z`(0,4))Zj(P,0) PdPd4) = 8^j•

00

Furthermore, the integral of each polynomial over the unit circle is 0, except for the first term, for which the integral evaluates to 1:

J I f/ 21c 1

J

r

1

Zi (p, 4)) pdpd4) = 0

00

if j = 1, otherwise.

For an aberration function expressed in terms of the Zernike polynomials, _ ^cili, where ci are the coefficients,

(7.13)

the average phase of the wavefront relative to a spherical wavefront at the pupil (the wavefront W in Fig. 7.1) is Cl, and the root-mean-square (rms) deformation is (see Exercise 7.2)

c^ .

A(Drms =

(7.14)

>1

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139

Zemike polynomials

If the aberration is small, according to Eq. (7.8), the normalized intensity at the geometrical image point is 1(Po) = 1 4^2

c.

(7.15)

j>1

Equation (7.15) implies that for a system suffering from a certain aberration, it is possible to introduce lower-order aberrations deliberately to increase the intensity at the geometrical image point. Suppose, for example, that an optical instrument suffers from spherical aberration ( D(p'$)

= CP 4 •

In terms of the Zernike polynomials, this aberration function is (j5,4) = 6 C Z11(P, $) + cP 2 — = 6 c Z11(15 ,0)+ 2 C Z4(P^^)+3Z1(P,l)•

We can introduce a defocus of C4 = -

to increase the normalized intensity from (1-4c 2 /45) to (1 — c2 / 180). Such deliberate introduction of lower-order aberrations is called balancing. Let us consider a balanced aberration function Iii that combines a single aberration term cp(p 72 , m4) and the corresponding lower-order balancing terms. Since the Zernike polynomials are a complete set, I a1 can be represented by the Zernike series

al



CJZj,

(7.16)

j=1

in which highest-order term Zj is the same as the single aberration term cp(pn, m4). By Eq. (7.15) the normalized intensity at the geometrical image point is i

i(Po) =1-4,n 2 1c^. j>1

Since c is positive semi-definite, the intensity i(Po) attains its maximum value if the series in Eq. (7.16) has the single term Zj, implying Obal = cjZj. The Zernike polynomials are naturally balanced. Each polynomial combines terms in such a way that the normalized intensity at the geometrical image point is a Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms

140

Aberrations

maximum. For example, spherical aberration is balanced by defocus in Z11, and tilt balances coma in Z7 and Z8. Besides the representation of Eq. (7.10), Zernike polynomials can be expressed according to the Fringe Zernike convention

Z^ (0, 0) = Rn (P)Yj ) • The coefficients of an aberration function expanded in terms of the Fringe Zernike set

c(0,0) = Y, cj, ZZ i

are related to those of Eq. (7.13) by c,=clan. Regardless of convention, an aberration function is in general an infinite sum of the Zernike set. In practice, the series is often truncated after J terms such that i

i

4)(p, 0) = 1 c1Z1 _ Y, c^Z^. i =1

(7.17)

1=1

The number of terms is usually 37 in photolithography applications. More terms are necessary in well-corrected (namely, low-aberration-level) systems where higherorder terms have similar magnitudes to the lower 37 terms.

7.4 Effects on imaging Although the exact impact of aberrations is specific to the object shape and illumination configuration, it is possible to describe the general effects of each Zernike term. In addition to the displacement theorem of Eq. (7.6), the concept of ray aberration is useful for our discussion. Consider an arbitrary imaging ray, say QQ'Pr P, in Fig. 7.1. It intersects the image plane at Pr = (x r , yr ), a point that is, in gen (xo, yo). The displacement of Pr from Po is the ray aberra--eral,difntomP= tion [51]: (Ax, Ay) = (xr

—

xo,Yr

—

Yo)

( a XcI aXcI In terms of the normalized variables (A9) = N o (AXI DY) _ NA RX( J: a(D ko R sin 8obi

51 'i •

ac:I: ac = afag

(7.18) )

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Effects on imaging



141

Piston

The first Zernike term Zl (p, 4)) = 1 represents the average phase of the wavefront; it does not affect the image.

Tilt The ray aberration corresponding to a wave aberration represented by = c2Z2 +c3Z3 = c225 cos4)+c32p sin0 is (A ,A9) _ ( 2 c2, 2 C3)The intensity distribution is shifted; but the image is otherwise unaffected. From Eq. (7.6), the amount of displacement is (&,A94) = ( 2 c2, 2 c3, 0 ). Defocus

With defocus aberration ( D(P)0)

= C4Z4 = C4V(2p 2 — 1),

the intensity distribution is unchanged except for a translation by (M,Af,OA) = ( 0,0,4Vc4).

(7.19)

Astigmatism

The effect of the aberration 'D(0 0) = C6Z6 = c« / 1

2 cos 24)

depends on the orientation of the object. A pattern varying only in the x- direction [such as that described by Eq. (5.5)] under coherent illumination with f(f,, g) = S(f , g) means that 0 = 0 or 0 _ Ir, so that (5, 4)) = c6 2 . This is a defocus. A pattern that varies in the 9-direction has a spectrum such that 4) = ir/2 or $ = 3rc/2, and c1(p, 0) = — c6\./p 2 . This is a defocus of the same magnitude in the opposite direction. Features that orient in the . - and 9-directions are focused onto different planes. The other astigmatism term

vp

0

(p, 0) = c5Z5 = c5V op t sin 20

affects patterns oriented at 45 deg and 135 deg with respect to the x-axis. In the presence of Z5 and Z6, different parts of an object experience different amounts of defocusing depending on their orientations. Illustrated in Fig. 7.2, a square opening having an elliptical image is one manifestation of astigmatism. Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms

Aberrations

142

Figure 7.2: The image of a square pattern [Eq. (5.26)] with d = 0.4, tf g = 1, and tb g = 0 under the influence of astigmatism (c6 = 0.1).

Coma The ray aberration corresponding to Z8 = c8 \(3p 3 — 2p) cos 4

is (Al, A9) = C8

/(9J2 + 3

g2 — 2 , 6.Jß).

For a pattern that varies only in the z-direction with J(f g) = 6(f, g), ( A9) =c8\(9fß -2,0).

While the —c82v term describes a constant translation (x-tilt), the c89^f^ factor represents an image shift that varies with the location of the light ray in the pupil, namely, the amount of displacement is a function of the object spectrum. In addition, the f^ dependence means that a ray at —Jo is shifted by the same amount as that at +fo. The image of a symmetrical object generally becomes asymmetrical. Such lateral asymmetry is orientation-dependent in the presence of Z7 and Z8. Nevertheless, the intensity distribution is symmetrical with respect to the z = 0 plane because of the absence of even power terms in p. Shamrock

Similar to coma, the Zernike polynomials Z 9 and Z 10 result in lateral image asymmetries. The 34 dependence means that patterns possessing a threefold rotational symmetry are the most prone.

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Measurement



143

Figure 7.3: In the presence of spherical aberration, light rays from the pupil intersect

the optic axis at different points. Spherical aberration A wavefront with spherical aberration [Z11 — 6p 2 + 1) and higher-order polynomials in which YJ (cß) = 1] can be perceived as having a radius of curvature that varies as a function of the radial position in the pupil. An image is degraded because light rays from the pupil intersect the optic axis at different points, as depicted in Fig. 7.3. Such degradation is asymmetric with respect to the Z = 0 plane, although no lateral asymmetry is introduced. High-order polynomials The effects of high-order terms can be ascertained qualitatively similar to those considered above. Light is diverted over a larger area with increasing n values because the ray aberration increases; the diffraction pattern spreads farther. Objects with mo -fold rotational symmetry are the most susceptible to Zernike polynomials with m = mo. Terms in which m is odd result in lateral but not longitudinal asymmetry.

7.5 Measurement The ability to measure aberration is indispensable for the production of high-quality optical elements. Knowledge of a system's aberration is beneficial to its users as it allows them to ascertain its performance and to monitor possible system degradation. There are three categories of aberration measurement techniques. One determines the aberration from the interference pattern formed by the wavefront of interest and a reference wavefront. Another class of techniques deduces the wave aberration from a set of intensity distributions produced by the imaging system in

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144



Aberrations

Light source

U

ice

Figure 7.4: Principle of the Michelson interferometer.

question. The third category measures ray aberrations from which wave aberrations are derived.

7.5.1 Interferometry Since we cannot observe phase difference by itself, we need to convert the optical path deviation into an observable quantity. Interferometry determines aberration by interfering the wavefront of interest with a reference wavefront. Phase information can be determined from the resulting fringes. There are many interferometry techniques."' As an example, we study the Michelson interferometer, whose principle is illustrated in Fig. 7.4. A quasi-monochromatic light source is collimated by lens Ll such that a plane wavefront impinges onto the glass plate P with a beamsplitting surface S. The fraction of the incident wave that transmits through S is reflected by mirror M1 and then by S, forming the reference wavefront W. The test wavefront is created by the portion of incident wave that is reflected from S. This light travels through the lens under test (L), being affected by aberrations that may be present, before reflecting from mirror M2 and transmitting through S, resulting in the test wavefront W'. The fringes produced by interference between W and W' can be observed through lens L2.

7.5.2 The extended Nijboer Zernike approach -

The aberration signature of a lens set is usually obtained using interferometry prior to system assembly and shipment. But this signature may not represent the aberra"'Malacara [52] provides a catalog of interferometry techniques and descriptions of their principles.

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Measurement

145

tion of the optical system because a final adjustment is usually performed at installation. It is desirable to measure the aberration of the installed optical system and to track its evolution. When in-situ interferometry [53,54] is unavailable, we need to determine aberrations by less direct means. One approach is the extended Nijboer-Zernike method [55]. In this technique, the through-focus coherent transfer function is represented by a linear combination of basis functions expressed in Bessel series. The coefficients of the basis functions are the same as the Fringe Zernike coefficients representing the pupil function. These coefficients are estimated by matching the theoretical and measured intensity distribution through focus. For an aberration function described by

(D(P, 0) =1 c Rn (P) cosm4, the field resulting from a point object is U(P,2) = 2 UÓ (P, 2)+ 2 ^i

m+l cJ n

(P1Z)cosm4,

j

where

n

V (P,2) = e

+i2 (-2i2)u-1 q buvJin+u+2v(2TCP)' u=1

v=0

u(27EP)u

(-1)g • (m+u+2v) • (m+v+u— 1)!(v+u-1)!(p+v)!u! 1 v + v + u)! u-1 v.)!v-}-u— q — )!(q — )!(p buv ( m+v)!( !( 2=kz(1— 1—sine 8 obi),

n+m p= 2 n—m q= 2 and the relationship between j, n, and in are given by Eqs. (7.11) and (7.12). For low levels of aberration, the intensity distribution is approximately I( P,4,2) = 4 IVó (P,2)I 2 + 8 1 c,Re {im +1vó * (P, 2 )VV (P,2)}cosmi j

=x8(P,z) (7.20) j

where +1UÓ (P 2 X (P>z) = f,Re {i ) V,, (P,Z)1, *

n

_ 4 8

ifn=m=0, otherwise.

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Aberrations

146

We need to estimate the coefficients c'j given a set of measured intensity distributions at various focus levels 2. Performing a Fourier analysis of the observed intensity distributions Imes (p, 2) with respect to the angular variable gives 21c

Wmea(P, 2 ) =

1

f Imea(p,^^z)cosm0Ci0.

(7.21)

0

Comparing Eq. (7.20) and Eq. (7.21) we obtain, form 0, the relationship Wmea(p, 2) — n

c'j (ß52).

(7.22)

Let us define an inner product

Po (Ymea,

X) = f Wmea(0, 2)X * (P, 2) PdO, z

0

with po being the range of p measurements and the summation covers all measured focus levels. The product of Eq. (7.22) with xn (p, 2) is Cj(in íX n') .

1 (mea,X )=

,

(7.23)

n

If we truncate the infinite series in Eq. (7.22) to N terms, and take the inner product of yrmea with the same N xn (p, 2) terms, Eq. (7.23) results in an N x N linear system

of equations. Its solution gives the least square approximation of the Fringe Zernike coefficients.

7.5.3 The Hartmann test The Hartmann test measures the relative shift of an array of imaged features [56, 57J. The ray aberration and subsequently the wave aberration can be deduced from the displacement data. Figure 7.5 illustrates the principle of the technique. The object contains an array of small openings, two of which are denoted by Ql and Q2 with their Gaussian images at Pl and P2. A screen S with a small aperture is inserted between the object and the first lens, such that the image of Ql (denoted by Pl) is formed by a small light bundle located at (a, b) in the pupil. According to Eq. (7.18), the displacement of Pl from Pl is proportional to the ray aberration at (a, b). Similarly, the shift of the image of Q2 (labeled P2) from its Gaussian image point P2 is proportional to the ray aberration at a different location of the pupil. Measuring the image displacements of all array openings allows us to sample the ray aberration over the pupil.

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Measurement



147

plane Figure 7.5: Principle of the Hartmann technique.

To derive the wave aberration from the measured ray aberration, we assume that the aberration can be represented by the first N Zernike polynomials N

c1Z1(0 0).

_

,

i =1

The measured displacement and the Zernike coefficients are then related by N

(Axmea,Aymea) _

az;

N

aZ1 1

C^—, Y, C^ á g ^1 af =i

J

.

For an object array comprising M openings, the Above relation becomes the matrix equation Omea = Ac,

where A is a 2M x N matrix in which each entry is the gradient of a Zernike polynomial, c is the vector of Zernike coefficients, and Omea represents the measured dis -placemntd.Thisrxquaoncbeslvdigthaquremod.

7.5.4 Aberration monitor patterns For optical systems with low aberration levels the phase factor exp(+i2itc) can be approximated by the first two terms of its Taylor series expansion: e +i 2

1c4D(P,O) = 1 + i2t(D(P, 0) = 1 + i2i I c1Zj (0, 0) -

With an aberration monitor pattern [58] described by Ok(x,Y) = Zk(j,g)e 12 +g 2 <1

ff

-i2T(fz+gy)

djdg,

(7.24)

the associated spectrum is

6k(P,$)

= Zk(P70)

for p < 1,

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Aberrations

148

and the scalar image field at the center of the pattern [(i, y) _ (0,0)1 under coherent on-axis illumination is, according to Eq. (4.39), 2n 1

11 Zk(P,0)e+i2nEcjZj(P,0)pdpdo Uk(0,0) ßt

00 2it 1

(7.25)

ffzk()[1 +i2itcjZj()] Pdodo 00

= i27tCk,

where we have neglected the obliquity factor for exposition convenience. The image intensity increases quadratically with the Zernike coefficient ck. By observing the center point intensity of objects designed for different Zernike terms k [using Eq. (7.24)], we can determine the set of Zernike coefficients that describes the aberration function of an optical system. In practice, it is difficult to accurately measure the intensity corresponding to Eq. (7.25) because ck is small. We can increase the intensity by adding a pinhole of transmittance p and phase O p to the center of the aberration monitor pattern [59], such that its transmission function and spectrum become Ok(x,Y) = p . e

g(x,9) +

ff Zk(f,

g)e-12Tc(.fx+gy)

dfdg

f^+g 2 1

and for < 1.

6k() =p . e + ` OP +Zk(P,c)

The image field at (z,y) = (0, 0) under coherent on-axis illumination becomes 2n 1

Uk(0,0)

ff [p-e+`OP+Zk(P,4)]

e+i2nEc;Z;(P ,4)Pdpd4

00

2n 1

JJ

[p - e + ` OP +Zk(P,0)] [ 1 +i2 n1c1Z^(0, 0)] Pdodo

00

= p , e +tO P + i27tCk.

If 4 p = it/2, the fields arising from the pinhole (p • e+ iOp) and from the aberration monitor pattern (i21ck) add in amplitude at the center point, resulting in an intensity of

Ik(0, 0) =p 2 +41tpck+47t 2 ck p 2 + 41t p ck = P 2 (1 +

47tck l

(

if p » ck.

p J

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7.26 )

Exercises



149

The intensity increases linearly with ck. Changes in the Zernike coefficient is transformed into changes in intensity. The principle of these aberration monitor patterns with pinholes is analogous to that of the phase-contrast method [49, 60].

Exercises 7.1 Derive Eq. (7.2). 7.2 Derive Eq. (7.14). 7.3 In the limit of small aberrations, show that the normalized intensity at the diffraction focus is 1— 4ík 2 c. j>4

7.4 A dielectric slab of refractive index n and thickness d is inserted between a spherical wavefront and its geometrical image point. Derive the equivalent aberration introduced by the slab. 7.5 What is the intensity perturbation at the center of an aberration monitor pattern if the pinhole transmittance is p = 0.5 and the Zernike coefficient is Ck = 0.01?

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Chapter 8

Numerical Computation The theoretical developments in the last chapters have provided us with equations that describe optical imaging in photolithography. With these equations, images of objects can be computed in lieu of carrying out exposures. The possibility to simulate allows us to harness the power of affordable computers to predict images of object patterns, and to optimize the object and exposure configuration given a desired image. In this chapter we discuss common numerical formulations for imaging simulation.

8.1 Imaging equations Summarizing results of previous chapters, the imaging equation for a system with lateral magnification M is [from Eqs. (6.14) and (6.68)]:

I(x,y,z)=K f ... f J(.f,k)

7

(f+.f',g+g)H * (.f+.f",g+g')

6(.f' ,g' )M(ƒ+f' , +g' )E0 . O* (.Í" ,g")M* (ƒ+.f" ,g+$" )EÓ

e - i 2 "9 d f dgd f'dg d f"dg'

=Kƒ ... f Jj.f,g)H7 (J+j+g )H * (.f +.P',g+g") , Muk(f +f',b'+8')Myk(f +.f„ b'+g")E^E^

i={x,y}

j{x,y} k={x,y,z}

6(f', g') Ô (J", g")e

-i2..p

d.Ídgd.f'dg d.f"dg', (8.1)

151

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152



Numerical Computation

where

K=

nimage

for coupling image [see Eq. (6.14)],

nP

for resist image [see Eq. (6.68)],

is refractive index of image space, is real part of resist refractive index,

nimage

nP

H

i

(4.31)

is the effective source,

J(f,g) -

ni a

ge 2

e +^2^^(P,^)

if Pe < sin8obi,

(6.69)

otherwise,

0

(7.10)

0(010) = 1 cizi (P, 0), j=1

for coupling image [see Eq. (6.14)], ge [ ^ see E .6. 68], for res i s ta f ii mae l )

_ Mo (f , 9) M(f ' g) M s tack(,Í^g)

(

ro'^' M0yx MO =

Px.,+Pxllx

PYj+PYIIx

oxy

MOYY = Px.Ly+PxIIY PY1Y+PYIIY oz= M0 PXIIZ PYliz

ß2 Px

aß

1_ y2'

__

(6.13)

aß

Px1y 1-72'

Prix=

-

1-12 ,

a2

PyIy = 1—Y2 ,

(6.10)

Py =O,

Px1z = 0 ,

_ a2

Pxllx — 1-721

Y

PYllx = 1aß 2 2

aß7 PxIIY = 1— y2'

PY IY

Pxllz = al

PYllz = ß,

1—y2 '

(6.12)

—

—

MS. Msyx

Mstack =

Ms

Msxz MSY, F±Px +Fjy Pxllx FLPylx+F xyPYllx

(6.66)

= FIPxly+Fl P lly FL PY 1Y+F PYIIY

Fi Pxll z

Fj PYllz

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Transmission cross-coefficient integration

153

F1 (p o z) _ t stack 1 re+ikP(dp—z)+p1bstacke—ikp(dp—z)1 Z substack L

(6.63)

1

,stack

11 (Po,z)

^Ikubstack [e

+ikP(dp _z) — P llubstack e —ikf (dp—z)1

(6.64)

,stack F^ ( Po ,z ) = tis^ubstack [e+' Pi

z (dp —z) +

1,

Pbstacke—ikP(dP—z)]

(6.65)

(6.59)—(6.62)

pij,zij,

dp is the photoresist thickness, = f sin9 obj,

ß=gsin8obj, y= 1—(f2 +g2 )sin e 8abj= pe = psinO obj,

= f2 + g2 , (P= (j'- 1")x+(S — g^)ƒ+(kp r— kp

„ )(z —

zo).

For computation of the coupling image, kp is the z-component of the wave vector in the coupling medium, and zo is zero. If our interest is in the resist image, kP is the z-component of the wave vector in the photoresist and zo is geometrical focal plane relative to the top of the photoresist. In writing Eq. (8.1), we assumed that • the spectrum is independent of the source point except for a translation proportional to the location of the effective source point [Eq. (4.40)]; • the diffracted field arising from each source point is proportional to the object spectrum at the appropriate spatial frequencies (f, g); • polarization of the diffracted field is independent of the object and the effective source; and • the ratio of the perpendicular to the parallel components is preserved during diffraction and propagation through the imaging system.

8.2 Transmission cross-coefficient integration To calculate image intensity at any point, we need to evaluate the sixfold integral of Eq (8.1), a daunting computation task. We can reduce the effort by assuming that the object is periodic in both the z- and 9-directions with periods p x and py . The spectrum then comprises discrete diffraction orders (see Exercise 8.1):

ó(M)=Er 5 (f — m ,g — n )O mn. n m

Px

Py

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(8.2)

Numerical Computation

154

An isolated pattern is approximated by a periodic object with a large period. Such an approximation is acceptable, except for coherent illumination, because the complex degree of coherence is vanishingly small if the separation between points is large. The image of an isolated pattern under coherent illumination can be computed using Eq. (4.39). With a discrete spectrum, Eq. (8.1) becomes an equation with a double integral: — i2ncp Om n O* m„ n„ e

Î(,9,z) = K

=

n!"n',m'

fff(.f,g)H(.f+m'/ß ,g+n/ßy)JJ*(. +fn"/p,g+n"/ß ) Mik (f', g )Mk (J", g")EiE dfdg (8.3) t={X1'}

k= {.r _ .Z}

Om'n, O n,^ nn e^i2' r TCCm of ;mrrn ri. z EiEY ,

K

j={.rp}

rt 1 .m^

k-{.r,y,z}

where +00

TCC W ;m„ n,,.Z =

LI

J(ƒ,g)H(J+m'/ß, + n'/ß)

H* (f + m"/ß, + n"/Py)Mik (5; z)MYk (Pë; z) d f dg. (8.4) Illumination in optical lithography is usually unpolarized. Since unpolarized light of intensity IQ is equivalent to two independent linearly polarized waves of amplitude IQ/2 vibrating in mutually perpendicular directions (see §6.1), I (x,Y, z) — 2 K

Om'n' Omnnn e —i2 '

TCCn„.mnnn. Z.

(8.5)

k={r.y.z}

To determine the intensity at a particular focus level z', we need to calculate the various transmission cross-coefficients TCC n ;m and sum up contributions from all relevant diffraction orders according to Eq. (8.3) or Eq. (8.5) [42, 61]. A possible computation procedure is: IIn°Z'

1. Calculate the object spectrum Ómn by Fourier transformation of Ô(,9) [see Eq. (4.33)]. 2. Compute the set of transmission cross-coefficients TCCm n,;m°n,,;Z by numerical quadrature. For each coefficient, the region of integration is defined by the effective source J(f,g) and the two displaced pupils H(f+m'/px ,g+n'/py ) and H(f +m"/px ,g+n"/p) ) (see Fig. 4.8). Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms

Source points integration 155

3. Sum contributions from different polarization couplings [indexes i, j, and k in Eq. (8.3) or indexes i and k in Eq. (8.5)] and diffraction orders [indexes (m', n') and (m", n") in the equations]. Computation complexity of this transmission cross-coefficient integration method (also called Hopkins' approach [19]) depends on the number of diffraction orders. The range of m' and m" is proportional to px and the range of n' and n" to p y , meaning that the computation time tcompute scales asymptotically with the square of the object area A m = px x py : 2

tcompute °C (C1tTCC+C2tsum)Am,

where tTCC is the computation time of one transmission cross-coefficient and tsum is proportional to the time needed to perform the multiple summations in Eq. (8.3) or Eq. (8.5). Since each transmission cross-coefficient calculation involves a 2D quadrature that requires evaluation of the computation intensive functions Mik(OO;Z)Mjk(Pë;z) and H(f,g)

while ts„m involves only a few additions and multiplications, tTCC » is im

and

tcompute « tTCCAm

For calculating images produced by a specific imaging system with objects having the same periods pX and py , the transmission cross-coefficients can be precomputed and stored, thereby shortening the computation time such that tcompute °` tsumA 1

8.3 Source points integration Instead of performing numerical quadrature to compute transmission cross-coefficients, the source points integration approach (also called Abbe's method) transforms the sixfold integral of Eq. (8.1) into a fourfold integral, which can subsequently be approximated by a summation. Such transformation is possible because source points comprising the light source are mutually incoherent. Intensities produced by all source points aggregate to give the final image. This addition property allows us to express Eq. (8.1) in the following form:

Î(î,z) — ^^j(.Í,g) ^ ^H(i+f ,g+g^)^(Í^ g ) J '

2

(

M.Eo)e-^2n[f'x+g'y+kA^(z -zo)] dpdg dfdg. (8.6)

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156

Numerical Computation

The quantity delimited by the absolute value is the electric field, generalized from Eq. (4.36) for polarization effects, arising from a source point of unit strength located at (I, g). If we denote the square of this quantity by Ícon , we can rewrite Eq. (8.6) as +00

I(,9,z) = I. (f,g)Icon(f,g;z)dfdS,

where Icon(f,;z) = K ^^H(f +f^,g+g)O(f^^g )

(

M Eo e —i2rz [!'z+g 9+Oz ' (z—zo)] d .

)

f'dg 2

We can perceive the overall image as the sum of an infinite number of weighed coherent images Ícon, with each component image arising from an effective point source of intensity f(fs , gs ) located at (fs, gs). For a periodic object with a discrete spectrum described by Eq. (8.2), +00

LI

i i

i



i

H(f+f ,g+8)o(f ,g)(M'Eo)e

f'x+'y+

2

'(z—zo)]

df dg

• E e—i2 [mx/PX +ny/ny +kP (z —zo)]

y) O ^ mn ( _ I H(.f + m/ß, g+ n/ ß)

0)

m,n

and Icon =

KI l (f +m/px,b' +nl b )O y

mn (M

Eo)

m,n



e —i2n [in /Px+n9/Py+kP' (z—zo)

2 ]

. (8.7)

In numerical computation, we approximate the effective source function by a finite number of point sources:

j(f,9)=Jf(J,g)

(8.8) s

where a s is the effective strength of the discretized point source located at 6(f — fs, g — gs). The image becomes I(x,f,z) =1 aslcon(Ís,b's;z),

(8.9)

s

and a computation procedure for this source points integration method can be:

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157

Coherent decomposition

1. Calculate the object spectrum Ómn from Ô(1,9) [see Eq. (4.33)]. 2. Approximate the effective source J(f, g) by the discretized source JA(f, g) [see Eq. (8.8)]. 3. For each discrete source point, compute the component image according to Eq. (8.7). 4. Sum component images to obtain total intensity [see Eq. (8.9)]. For an effective source of area A and an object of area A m , the computation time scales according to S

tcompute — As . Am

8.4 Coherent decomposition We can also reduce the sixfold integral of Eq. (8.1) into a fourfold integral by (inverse) Fourier transformation of the constituting functions into their spatial-domain counterparts. For scalar imaging we essentially recover Eq. (4.32) from Eq. (4.35) (see also Exercise 4.4):

I(x,Y) =

Nl

w (x — zá,Y — yo,x — xö,Y — Yo) o(xo,yo)O' ( 1o,9o)d1od9d1ád9ö,

where

W (.óY10;x ool ,Yo) = f((xo xo^,Ya yo)H(xo,Yo)H * (xo,Yu)• —

—

The function W can be decomposed into a series of its eigenvectors cpk [62-64]: W (xo}Yo;x11 40) _ ^k(Pkl-oJo1 ) (Pk(xoJo} k=1

where Xk is the eigenvalue corresponding to the eigenvector cpk. With this series representation, the image becomes

I(19)—k^kkffff (Pk(x—z Y—Y)Ô(xY) (P*(x—.x',9-9")O*(xß,3 ")d2'd9' dx"dy"

_ k=1



ff (Pk(.x-x,9-f)O(x,Y)d 'dy

+°°

1

,

2

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Numerical Computation

158

We can interpret each eigenvector as the transfer function of a coherent imaging system [compare with Eq. (4.1)]. The overall image is the weighed sum of the images produced by an infinite number of coherent systems. In situations where the magnitudes of a few eigenvalues are much larger than the rest, we can approximate the image as a finite sum of these dominant eigenvectors: K

1(1,9) =

2

(8.10)

kk (O ® Pk) (x, Y) I , (

k=1

where K is the number of significant eigenvectors, and the symbol ® denotes convolution. To calculate the image intensity at any point, we need to compute K 2D convolutions. Equation (8.10) is amenable to efficient computation of objects comprising polygons. For example, a rectangular pattern of foreground transmittance tf g and background transmittance tb g = 0 with vertices at (ui , 9i), (12, 91), (12, y2), and (11,92) can be described by O(x,Y) =tfg[Q( 1— zl,9 - 9l) — Q( 1-12,9 - 91)+

Q(1-12,9-92) — Q(x — x1) — y2)], where 1 Q(x ' Y) 0

ifx>0and9>0, otherwise,

is the quadrant function. An object consisting of N rectangles can be described by

N

o(x,Y) _ tfg ) [Q(x -1 1n) ,y — Yins ) — Q(.x — x2 'Y — Yins )+ n=1

Q(x—x2n^'Y—Y2n)) — Q(x — x1n ^,Y — Y2

where the superscript (n) indexes the rectangular shapes. Substituting this object description into Eq. (8.10) results in I(x,Y) = I Xk ( 6 ®^k)(xiY) 2 k=1 K IN

(n) (n) (n) ^ (n) 1vk(x—x1 ,9 91 ) — lk(X -12 'YY1 )+

_ ^?k k=1

-

I n=1 2 (n) (n) (n) (n) 1 1Vk(x —.z2 ,Y — Y2 ) — Mfk(x — x 1 ,y—y 2 ) J , (8.11)

where 11k(1,9) =

Q(x,y) ®(Pk(1,Y)

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Object spectrum

159

is the convolution of the quadrant function with the kt eigenvector. As the functions 1rk(1,9) are independent of the object, they can be precomputed (for each particular exposure configuration). Image calculation becomes a double summation over the significant eigenvectors [index k in Eq. (8.11)] and over the rectangular shapes [index n in Eq. (8.11)]. Each inner loop requires four function look-ups and three additions, and each outer loop needs two multiplications. The computation cost is tcompute = NK(4tlook-up + 3 tadd) + 2Ktmuit + (K —1)tadd [+Kty^], where tlook-ups tadd, tm„lt, and t, are respectively the time needed for function lookup, addition, multiplication, and computation of a convoluted function 111k (1,9). Assuming the functions yrk(1,9) are calculated in advance, the computation time scales linearly with the number of object vertices asymptotically. The procedure for image computation by the coherent decomposition approach is:

1. Determine the eigensystem of W, and the number of retained eigenvectors K. 2. Precompute the functions yrk (x, y) for all significant eigenvectors. 3. Compute the image according to Eq. (8.11).

8.5 Object spectrum The object spectrum is the Fourier transform of the field transmitted by the object under the illumination of a unit-amplitude wave. This field behaves in a complicated manner, varying continuously and oscillating in the vicinity of transitions between regions of different transmittance. To simplify image computation, we make the approximation, similar to Kirchhoff's in his derivation of the diffraction integral [Eq. (3.7)], that the transmitted field changes abruptly according to the transmittance of the object. This thin-object approximation (also called thin-mask approximation), of which Eqs. (5.5) and (5.26) are examples, suffices for patterns sizes large compared with the wavelength. When the object dimensions are on the order of or smaller than the wavelength, we should treat object transmission as a boundary value problem based on Maxwell's equations, and solve for the transmitted fields numerically. For example, for the space depicted in Fig. 8.1, we need to find the field across the line AB subject to an incident radiation. The computation should consider material properties, including the refractive indexes of the mask substrate nglass, the opaque layer n opa , and the object space medium nobject, as well as the thickness d opa of the light-blocking material.

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Numerical Computation

160

Figure 8.1: The field along AB can be determined by solving Maxwell's equations. The thin-mask approximation simplifies object spectrum computation.

Figure 8.2: Time-domain finite-difference is one approach for computing the field along AB in Fig. 8.1.

Time-domain finite-difference is one approach for solving such a numerical problem [65-68].' Using the integral form of Maxwell's equations [Eq. (1.6)], we solve for the six electromagnetic field components

e = ( ex

,

ey , ez )

and

h = (hX hy , hZ ,

)

by decomposing the object into a grid and assigning one field component to each grid point, as illustrated in Fig. 8.2. For a cubic grid in which Ax = Ay = Az, the surface and line integrals in Eq. (1.6) are evaluated on squares. Referring to Fig. 8.3, the z-component of the electric field ez(iAx,joy,kAz) = ez(i,j,k) =e(,,)

can be expressed in terms of the x- and y-components of the magnetic field:

á

(£e t +óc)ez(i,j,k)

= [h,(i,j— 1/2,k) —h,(i, j+ 1/2,k)+ by (i+1/2,j,k)—by (i-1/2,j,k)]Ax. (8.12)

'Besides the time-domain finite-difference approach, the boundary value problem can be solved by finite-element [69] and frequency-domain methods [70-75].

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Object spectrum



161

hX (i,j+1 /2,k) Ay/2 b y (i -1 /2,j,k)—e, k)—h y (i +1 /2,j,k) Y hJJ-1/2,k)

ZLX

Ax/2

Figure 8.3: The z-component of the electric field is expressed in terms of the x- and y-components of the magnetic field.

This equation is derived under the assumption that ez (i, j, k) is constant over the square surface and that the magnetic field components are constant along each line segment comprising the square. To obtain a time discretization of Eq. (8.12), we assume that the electric field components are constant within the time period [nAt, (n + 1)At), and that the magnetic field components are fixed within the duration [(n — 1/2)At, (n + 1/2)At). This formulation results in the time-domain finite-difference equation: eZ

+1(> >) =

aez (, ,) + ß [hx +1/2 (, —1 j2, ) — hz + 1/2

where

_

a

(,1/2,) +hy

+i/2 ( 1 / 2

,,) — by

+i /2 (

-1 / 2

,,)11 (8.13)

2Ee — G At _ At 2 and 2£e + 6cAt ß dx 2Ee + 6,Ot

Following the same procedure, the equations for the other five components are ey +1 ( 1/2,1/2)= aey(,1 /2,1/2) +ß[hz +i /2 (-1/2,1/2,1/2)— h 1/2 (1/2 1/2,1/2) +h 112 ( 1/2,1) —hX+z /2(,1/2,)], (8.14)

e

x +l (1 /2„ 1 /2) = aex (1 /2„ 1 /2) + [hy+i

/2 (1 / 2

„) —

by +1/2 ( 1/2 1) +hz +i /2 (1/2,1/2,1/2) —hz +1/2 (1/2,-1/2,1/2)1, (8.15) hz +1/2(1/2 1/2,1/2) = hz -1 / 2 (1/2 1/2,1/2)—

At1 [e,'(1/2„ 1/2) —eX(1/2,1, l/2)+ey(1,1/2,1/2) —ey(,1 /2,1/2)], (8.16) Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms

Numerical Computation

162

n-1/2

n+ l2

by

(1/2 )=hy l „ At 1

hx+l/2 (,

(1/2,,)—

[e?(„)—ez(1„)+ey(1/2„1/2)—ey(1/2„-1/2),, (8.17)

1 / 2 ,) = hX—' 12 (^ 1/2,)— At

I ez(,1,)—ez(„),. (8.18)

The field transmitted by the object can be computed by iteration of the above equations. To confine the computation volume, absorbing boundary conditions are needed on the border surfaces of the simulation domain. These conditions result in a different set of iteration equations compared with Eqs. (8.13)—(8.18). Please refer to the literature [76] [77] for details. Electromagnetic simulation using the time-domain finite-difference approach thus comprises two primary steps: 1. Discretize simulation volume; assign (material) parameters to each grid point. 2. Iterate field components according to Eqs. (8.13)—(8.18) or boundary conditions until convergence.

8.6 Remarks The formulation of Eq. (8.1) assumes that the wafer stack is a stratified medium. In situations where there is significant wafer topography effects, the analysis of §6.4 is inadequate. There is generally no close-form solution. The resist image must be computed rigorously by consideration of light coupling into the photoresist in the presence of wafer dielectrics [78, 79]. The time-domain finite-difference technique described in §8.5 is one method for such calculation. Besides images in photoresists, we are often interested in modeling photoresist dissolution. The effects of resist processing can be simulated by two classes of techniques. Rigorous approaches [80-84] attempt to describe the photoresist dissolution process with first-principle formulation involving reaction and kinetics of chemical species within the resist. Phenomenological methods [85-88], on the other hand, seek to model photoresist behaviors heuristically by calibrating actual resist performance with model parameters that may bear little relationship with resist chemistry and physics. Interested readers may wish to consult these references for details.

Exercises 8.1 Show that the spectrum of a periodic object with periods px and py in the i- and 9-directions is given by Eq. (8.2).

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Exercises



163

8.2 Derive the spectrum of a periodic rectangular feature with periods pX and py in the x- and 9-directions. The coordinates of the lower left and upper right corners of one of the patterns are (io,9o) and (xl,yl). Each rectangle has a transmittance of tfg with the background transmittance being tbg . 8.3 Is it possible to simulate immersion imaging with a simulator that assumes the refractive index of the coupling medium to be one? If so, how? 8.4 Using Stoke's theorem (see Exercise 1.5), develop a formula to compute the area of a polygon.

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Chapter 9

Variabilities In replicating an integrated circuit layout during fabrication, the same object shapes are often delineated numerous times. Because of unavoidable variabilities of a manufacturing process, the delineated shapes are generally different from the nominal shapes and from one another. This variability should be kept under the specified tolerance according to which integrated circuits are designed. Too much deviation causes circuit failure. An understanding of the various causes of variation is helpful in devising means to reduce, stabilize, and compensate for the undesirable variation. Although all processing steps (such as deposition, lithography, etching, and chemical-mechanical polishing) contribute to patterning nonuniformity, we focus on variabilities arising from optical imaging, since, with both layout shapes and image tolerance shrinking rapidly compared with Xo/NA, control of image variabilities is of increasing concern. Lithography becomes more difficult with decreasing kl and k1h_P;tch.

9.1 Categorization We can classify the causes of lithography variability into two categories. One affects object shapes located in identical environments, and the other impacts the same object shapes situated within distinct configurations of neighboring shapes. Let us call the effects of the former fluctuations and those of the latter inherent variations. The same cause can result in both fluctuation and inherent variation. On the other hand, a fluctuation and an inherent variation may have similar manifestations. To distinguish fluctuation from inherent variation we must first expound the meaning of environment. Let us define afeature as an object shape within a particular configuration of neighboring shapes. The shapes A and B in Fig. 9.1, although identical, are distinct features because their environments of shapes are different. The periodic space described by Eq. (5.5) is a feature that is fully specified by its foreground and background transmittance tf g and tb g , size d, and period ßx. A peri-

165

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Variabilities

166

Al

A2

A

gi

B B2 B2

Á

A

B^

l A2

Bj

Figure 9.1: A feature is an object shape within a particular configuration of neighboring shapes. A and A' are the same feature if we identify edge Al with Ai and A2 with A. odic line feature is similarly determined by these four parameters:

OX(i) = tbg if Ix—npx l < cl/2, n E Z, g otherwise.

ti

(9.1)

The orientation of a feature does not constitute its environment. For example, a y varying periodic line described by -

^

tb g if IY — nPy I < d/2, n E Z, y ^ y ^ — tfg otherwise,

is the same feature as that described by Eq. (9.1), provided they have the same tb g , tfg , and cl, and px = py . With this understanding of "feature," we can define fluctuation as the variability of the same feature, and inherent variation as the difference, excluding fluctuation, between distinct features .of the same object shape. Referring to Fig. 9.1, the difference between the delineated shapes of A and A' (and that between those of B and B') is fluctuation. So are differentiations between their delineated shapes across an exposure field, from wafer to wafer, and from lot to lot. The distinction between the averaged image of all replicas of A and A' (and those of B and B') is inherent variation. By this demarcation, we can perceive fluctuations as arising from engineering imperfection while inherent variations as consequences of the laws of physics. The variation in the images of periodic spaces with the same nominal dimension d but different periodicities follows from Eq. (4.35) because of differences in their spectra. But no physical law dictates that images of a feature at two points in the field of an optical instrument should differ. The difference one may observe can be caused by aberration fluctuations of the imaging system.

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Proximity effect



167

A

pthreshold

w

O

-1.0

-0.5

0.0

0.5

1.0

x (7JNA)

Figure 9.2: Method for determining the size and placement of a space image.

To quantify image variability, it is useful to convert each image into a number or a set of numbers that represents a relevant quality of the image. Metrics for this purpose include contrast, depth of focus, exposure latitude, normalized image logslope [89], exposure-defocus window [38], and total window [90]. These metrics can be determined from measured or simulated data. As an example, Fig. 9.2 illustrates a method that derives image size and placement from a computed image. The image of a feature, a periodic space in this case, is first computed using the techniques described in Chapter 8. The size iv of the image is the distance between the intersections of the simulated image and a threshold intensity 'threshold.' The placement error 0 can be defined as the distance between the center of the object shape and the midpoint of the two intersections. We use this method in our investigation of variabilities in subsequent sections of this chapter.

9.2 Proximity effect The spectrum of an object spreads as its size decreases. For dimensions on the order of or smaller than the wavelength, the pupil cuts off spatial frequencies that carry a sizable fraction of the light energy transmitted through the object. The image becomes distorted. One manifestation of optical distortion is proximity effect, an inherent variation where the image size w of a shape changes depending on its environment. A typical scenario is shown in Fig. 9.3, which plots the image sizes of nominally d = 0.6 spaces [Eq. (5.5)] and lines [Eq. (9.1)] as a function of period. As the period changes 'We can relate !threshold to the exposure dose Dprint that is supplied in an exposure. Denoting the dose to clear a positive resist or to harden a negative resist with a completely transmitting mask by Do, the dose D pri ❑t is Dprint

Do 'threshold

This approximation becomes more accurate with increasing photoresist contrast [85].

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168



Variabilities

❑--❑

space

Q o Z

0-0

.

line

N

N_ (0

O N ^ N

.1

O

1.0

2.0

3.0

4.0

period (RINA)

Figure 9.3: Proximity effect is the change in image size depending on the environ-

ment of an object shape. The features are nominally d = 0.6 spaces and lines with different periods.

the image sizes of periodic lines vary by as much as 20%. ü Figure 9.3 suggests that periodic spaces exhibit less proximity effect than periodic lines. We can use Eq. (4.32) to investigate whether this is generally true. Let us first contrast the imaging of two periodic spaces of the same size dbut different periods: pi = 2d and P2 = 4d. Imagine that the object is divided into atomic regions of width d, as illustrated in Fig. 9.4(a). The integral in Eq. (4.32) can be converted into a sum involving interaction of these atomic regions. For tfg = 1 and tb g = 0, the intensity at space Ao is roughly +00

I2d 0e ^ 1 J(0, 0)H(2nd, 0)uY* (2nó, 0) } n=--

I, l (2 [i — j]d, o) H(2id, 0)R* (2 jd, 0) (9.2) iii

and +00

,space f(0, 0)H(4nd, 0)H* (4nd, 0)+ n=--

J(4[i— j]d,0)H(4id,0)PI*(4jó,0). (9.3) i0i

Except for coherent illumination, the mutual intensity J(x,9) generally decreases with increasing 1.xJ and y^. For circular illumination with a partial coherence factor 6 [Eq. (5.13)], the first dark ring of f(î,9) occurs at [see Eq. (4.29)] r

_

z2 + y2

_ 0.61 6

"Proximity effect generally means any type of image shape distortion including pattern shortening and corner rounding. But it has the specific meaning of image size variation with period in this context.

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Proximity effect



169

2d 0

0

— — 1ine — —

0

0

0

O 0 0 0 0 0

Ap

A2

A4

A -1 A l

pl =2c1

pl = 2d

4d O

A3 A5

' line \ —

O

0

Ao

A4

00 000

OOO 00

A - ,A - ,A -1 A l A 2 A3

A5 A6

pl =4J

(a) space Figure 9.4:

(b) line

Spaces generally exhibit less proximity effect than lines.

If we assume 6= 0.8 and d=0.4, f(î,9)0

for r > 0.8 = 2d.

(9.4)

With the approximation of Eq. (9.4), the intensities expressed by Eqs. (9.2) and (9.3) become I2ppace.

+00 N

j(0, 0)H(2nd, 0)H* (2nd, 0),

and

n=--

Isppace

+00

N

f(0,0)hH(4nd0)H*(4nó,0). n=--

Since the point spread function H(z, y) generally decreases with increasing r, with the first zero at P = 0.61 for a circular aperture [Eq. (3.24)], only the first term is significant in both sums. The intensities of the two spaces are approximately equal. The situation is different for periodic lines. Dividing the object into atomic units in a similar manner [Fig. 9.4(b)], the image of a line of size d and period pl = 2d is roughly

72ó

e

—

If(o, 0)R( — [n2 + 1], 0)R ( — { n2 + l], ) +

n=--

ll,

([i— j]2d,0)H(— [i2+1]d,0)H*(—[j2+1]d,0).

i0i

The sum over i and j can again be neglected due to Eq. (9.4). But the double sum is non-negligible for a line of period p2 = 4d. For this feature, the atomic units such as A 1 and A2 are separated only by d. They interact and contribute to an image that can be much different from the image of a line of period pi = 2d. Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms

Variabilities

170

In addition to mask tone, proximity effect also depends on illumination configuration, mask technology, and wafer stack through their presence in Eq. (8.1). Although the exact behavior depends on the detailed interaction between the object spectrum and optical system [Eq. (5.1)], proximity effect generally increases with decreasing kl and k1h_p;«h factors, and with an increasing degree of illumination coherence. For example, the first zero of the complex degree of coherence under partially coherent illumination [Eq. (5.13)] is, according to Eq. (4.29), r=0.61 ko aNA The range of non-negligible optical interaction increases with decreasing partial coherence factor.

9.3 Object variabilities (photomask errors) 9.3.1 Dimensional error Photomask dimensional error is the deviation of a mask-feature size from the designed size. From a mask maker's viewpoint, dimensional error comprises both fluctuations and inherent variations. For the purpose of our discussion, all mask dimensional-error components are considered fluctuations. When an object pattern is large, its image is scaled exactly, except for a constant bias bo, by the magnification of the projection optics M. The image size wo varies linearly with the nominal mask dimension do with a slope equal to the magnification M: wo = Mdo + bo. Deviation of mask dimension from the nominal (Ad) results in a scaled linewidth error (Ow): Aw=w —wo = M(d — do) + (bo — bo) = MAd.

A unit change in the mask dimension corresponds to a unit change in the image size scaled by the magnification of the exposure system, a phenomenon illustrated in that portion of Fig. 9.5(a) where mask dimensions are large. The effect of mask dimensional errors is diminished in a reduction system. Image sensitivity to mask variation generally increases with decreasing feature size. When the object size is small, the effective mask dimensional error MAd is magnified [mask dimensions smaller than 0.5Xo/(NA x M) in Fig. 9.5(a)]. The severity of this amplification is described by the mask -error factor (MEF) [also

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Object variabilities (photomask errors)



171

U,

U,

0—o space

Z o

ó ^

N

w

0—❑

Ü

N N

square

Q

a)

N

^

E ^? 0

N

co

(C

E

1.0

0.5

1.5

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

d (GINA)

mask dimension [X/(NA x M)j

(b)

(a)

Figure 9.5: (a) Image size varies linearly with mask dimension for large features, but sensitivity to mask variation increases with decreasing features size. The features are periodic spaces with ßx = 2d [Eq. (5.7)]; and the magnification factor is assumed to be 0.25. (b) The mask-error factor typically increases with decreasing feature size, although the exact behavior depends on all aspects of the imaging process, including illumination configuration and mask technology. [From A. Wong, et al., "The mask error factor in optical lithography," IEEE Transactions on Semiconductor Manufacturing, vol. 13, no. 1, pp. 76-87 (Feb. 2000), copyright (2004) IEEE.]

called mask-error enhancement factor (MEEF)] [91, 92], such that the relation between mask dimensional error and image size fluctuation becomes Aw = MEF x MAd. As an example of mask-error factor dependence on object size cl, Fig. 9.5(b) plots MEF for the same spaces as those in Fig. 9.5(a) as well as for a square [Eq. (5.26)]. The MEF is I for large features, but increases from unity as d decreases below 0.75 for the square and 0.5 for the space. Because of its amplification for small object sizes, mask dimensional error is a significant fluctuation contributor in low-k1 optical lithography. i^i "'In addition to its dependence on object size, the mask error factor also depends on feature type and its environment, photomask technology (chromium-on-glass, attenuated phase-shifting mask, and alternating phase-shifting mask), and illumination configuration (on-axis and off-axis). In general, the factor increases with decreasing kl and k1half-P,«h. One exception is the imaging of a narrow, sparse line by alternating phase-shifting masks in which the MEF approaches zero with reduction of the chromium size [931.

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172



Variabilities

0

d=0.10 ?/NA

CR ó

---------------d=0.051JNA

>'(0 to o

N •_

t Ó

N O

-2

-1

0

1

2

x (AJNA)

Figure 9.6: The image intensity is quartered when the size of a small isolated space is halved [23].

Mask-error amplification arises from image integrity degradation. Consider a 1D isolated space of width d: _ 1

^x(x) 0

if 1 xß < J/2,

otherwise.

Its spectrum is ô(f) =dsinc(fd). In the limit of small d, lim Óx (f) = d, d-*O

and the image is [see Eq. (4.37)] I () d

ff

TCCX(t ; ") e -i21r [íf'-f")z1 d f' d J"

—

(9.5)

= d^L(J, H). The image shape is determined by L(J, H), a quantity sometimes called the line spread function. Since L(J, H) is a function of the effective source intensity dis J and the coherent transfer function H, the image shape depends only on-tribuon exposure system parameters. Modification of the object has no effect on the image shape, except for scaling of the intensity by the square of its size. Halving the size quarters the intensity. This phenomenon is illustrated in Fig. 9.6 for two spaces of sizes d= 0.1 and 0.05. The quadratic dependence of image intensity on size described by Eq. (9.5) is the primary physical cause of the mask error factor. We can qualitatively ascertain

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Object variabilities (photomask errors)



173

O

N Y c0 Q

Eq. (9.5) W) Eq. (4.38)

—

Ó .^ 6 0 O 0 d r

a) .^ 0 D)

O

0.0

0.2

0.4

0.6

0.8

1.0

d (XJNA)

Figure 9.7: Severity of mask error amplification can be ascertained by comparing the peak image intensity predicted by Eq. (4.37) and Eq. (9.5). [From A. Wong, et al., "The mask error factor in optical lithography," IEEE Transactions on Semiconductor Manufacturing, vol. 13, no. 1, pp. 76-87 (Feb. 2000), copyright (2004) IEEE.]

the severity of error amplification by comparing the square root of the actual image peak intensity with that of Eq. (9.5). A large difference means that mask errors are not amplified much, whereas closeness in the numbers indicate large effective magnification. Plotted in Fig. 9.7, the difference between the straight line, which represents Eq. (9.5), and the curve, which represents the actual image peak intensity, is big for large spaces. As the size decreases from d = 0.5, the curve and the line begins to converge. Below d = 0.2, changing the object no longer changes the image shape. Image control is lost. A similar consideration applies to squares with minor changes of the imaging equation (see Exercise 5.10):

I^x,9) _ [

yJ (r)

r

J

Z d4 •

In the limit of small cd, the image shape is solely determined by the point spread function while the intensity varies with the size. Rather than a quadratic dependence, the intensity is proportional to the fourth power of the linear dimension of the square contact hole. Because of this fourth power dependence, the MEF of contact holes starts to increase from unity at a larger dimension and has a higher value than that of spaces at the same dimension. The MEF can be as high as 4 for 0.5X0/NA contact holes, offsetting the advantage of 4X reduction systems. Although loss in coupling image integrity is the primary cause of the MEF, all factors causing image quality degradation, whether coupling, resist, or latent, worsens the MEF. Focus error, aberrations, excessive diffusion of chemical species in photoresists, and low photoresist contrast all increase the MEF.

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174



Variabilities

incident radiation

Figure 9.8: An alternating phase-shifting mask fabricated using a subtractive process.

9.3.2 Phase and transmission errors Phase and transmission errors are results of nonuniformities of phase-shifting mask dielectrics. They can arise from mask topography, imperfection of defect repair, and thickness nonuniformity of photomask films. As an example of phase and transmission errors, consider an alternating phaseshifting mask fabricated on a chromium-on-glass substrate using a subtractive process. Illustrated in Fig. 9.8, the 180 deg regions are etched areas of the fused silica substrate. The thickness of material removed detch is determined by requiring an optical path difference of

2n+1 ^

aEZ 2 (9.6)

between light passing through the 0deg and 180 deg phase regions. For a path difference of 2/2, a first-order estimation of the required etch depth detch is

xo

nglassdetch — nobjectdetch =

i

2nobject

detch =

(9.7)

2nobject(nglass — nobject)

where nglass is the refractive index of the fused silica substrate at the exposure wavelength and nobject is the refractive index of the object space. For wavelengths of interest in optical lithography (436, 365, 248, 193, and 157 nm),

nglass

=

1.5,

nobject = 1 ,

and detch = kO

The etch depth is approximately equal to the exposure wavelength. The phase regions are asymmetric because the 180 deg areas are etched while the 0 deg are not [94-96]. This imbalance causes transmission and phase errors. Figure 9.9 shows the simulated field within an alternating mask using the time-domain finite-difference technique described in §8.5. The incident light is reflected from Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms

Object variabilities (photomask errors)

175

Figure 9.9: The Odeg and 180 deg phase regions are asymmetric because of mask edge scattering [23]. the chromium areas, resulting in the standing waves. Transmitted fields through the two openings are not identical. The darker color in the unetched 0 deg region on the left compared with the etched 180 deg opening on the right suggests higher transmittance of the 0deg region. Detail analysis also reveals that the phase difference between the regions is not exactly 180 deg, even when the etch depth is that given by Eq. (9.7). The effects of phase and transmission errors on images are demonstrated in Fig. 9.10. For an alternating phase-shifting mask with only transmission but no phase error [Fig. 9.10(a)], the region with lower transmission gives a lower peak intensity for all focus levels. With only phase error [Fig. 9.10(b)], there is no intensity imbalance at nominal focus. The peak intensity of one opening is greater than the other with focus error in one direction, and the difference is reversed when the focus varies in the other direction. The effect of pure phase error is a focusdependent peak intensity difference. With a combination of transmission and phase errors [Fig. 9.10(c)], the peak intensities at best focus are not balanced because of transmission error. The imbalance is not maintained through focus due to phase error. With intensity imbalance, spaces that have the same nominal dimension print with different sizes in positive photoresists. Lines are shifted; but their sizes remain mostly unchanged. The effects are reversed in negative photoresists: resist islands differ in size while clear areas are displaced.

9.3.3 Edge roughness We mean by edge roughness the minute and random undulation of the contours of a photomask feature. By analyzing its spectrum, we can classify roughness into two categories: low-frequency unevenness that is resolved and replicated onto the wafer, and high-frequency coarseness where spatial frequencies are beyond resolution by the exposure configuration in use (on the order of 2NA/X0; see §5.6). Although these high-frequency components are not reproduced directly, they may collectively cause an effective mask dimensional error.

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176



Variabilities

0

k

ó

positive

focus

O

4 O

C N O

C N

r

d

Q)

-0.8 -0.4

0.0

0.4

-0.8 -0.4 0.0

0.8

x (a/NA)

0.4

0.8

x ()/NA)

40.00.40.8

x (AJNA)

Lo n ó

nominal focus

_T mus,

_T

C

'(/1 C

C N

C

W O d

0

m ó

d

C N ó

ó -0.8 -0.4

T

0.0

0.4

0.8 -0.4

0.8

x ()JNA)

x ()/NA)

0.0

0.4

0.8

0.4

0.8

x (AMA) N r ó

_T í1 O7

negative focus

^N O

C N

^N O

. O d N

C N

C N

ó

O

-0.8 -0.4

0.0

0.4

0.8

x ()/NA)

(a) transmission

-0.8 -0.4

0.0

0.4

0.8

-0.8 -0.4

x (A/NA)

(b) phase

0.0

x (2JNA)

(c) both

Figure 9.10: Effects of phase and transmission errors [23].

9.4 Polarization effects Not only does photomask topography cause phase and transmission errors, as demonstrated in §9.3.2, it also changes the degree of polarization of light. Let us begin by studying transmission of linearly-polarized light through an opening, as illustrated in Fig. 9.11(a). The incident light can be polarized with the electric field vibrating parallel or perpendicular to the edge of the opaque layer, which is assumed to be chromium. Figure 9.11(b) plots the fraction of energy transmitted through the opening normalized to its width for both polarizations, where TE (transverse electric) denotes electric field vibration parallel to the chromium edge, and TM (transverse magnetic) represents perpendicular vibration. The TM polarization shows higher transmission for all opening widths; the fractional difference increases with decreasing opening size [97]. Higher transmittance of the TM polarization may be traced to the electromagnetic boundary conditions at the air-chromium interface. Because the absorbing chromium layer is metallic-like, the electric field component parallel to the chromium edge is close to zero, whereas the perpendicular component is not (see

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Illumination



177

0 ^Q

TE O

TM

a,=248 nm

TE O —TM nglass ...:..:.::

C

cs

L

y

0

0

ir_

U

.

c chromium

N cs

0

250

500

750

1000

opening width (nm) (a)

(b)

Figure 9.11: (a) Structure of an opening. (b) Fraction of energy transmitted through the opening normalized to its width for the TE and TM polarizations [97]. Exercise 6.9). Unpolarized light becomes partially TM polarized upon transmission through photomasks. The degree of polarization increases with decreasing opening width. Since images of TM-polarized light have lower contrast than those of the TE polarization (see §6.7), changes in the degree of polarization degrades images by emphasizing the TM component [98]. Despite adversely affecting image quality, photomask-induced partial polarization does not cause inherent variation or fluctuation for unpolarized illumination. Under polarized illumination, however, the degree of polarization of transmitted light is a function of feature orientation. Variability would manifest as orientationdependent image shapes, convolving with the effects of astigmatism (see §7.4).

9.5 Illumination When the effective source intensity distribution is asymmetric with respect to the optic axis, the exposure system suffers from nontelecentricity. Since an asymmetric effective source implies a preponderance of light incident from some off-axis direction, we can understand the fluctuation arising from nontelecentricity by studying imaging with an off-axis point source, as illustrated by the schematics of Fig. 9.12. Imagine an image formed from a bundle of rays centered at the undiffracted direction indicated by the bold arrow in the figure. The lateral position of this center ray shifts with the longitudinal (focus) planes. Nontelecentricity causes image placement error that increases with defocusing.

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178



Variabilities

out of focus^ \i

best focus

Figure 9.12: The center image-forming ray shifts with focus for an off-axis source point [23].

A remedy for such focus-dependent image shift is to maintain source symmetry, such that the image shift corresponding to one source point is compensated by that of the mirror source, resulting in a final image that has no net displacement. This is the rationale behind dipole illumination (see §5.3). One can observe the effective source intensity distribution on the image plane using the method illustrated in Fig. 9.13 [99]. Rays from the light source impinge upon a clear reticle with an opaque spot P on the back. With Köhler illumination, the intensity on the object plane would have been uniform in the absence of the opaque spot. Let us denote this intensity by I. With the opaque spot, the radiance on the object plane diminishes. For example, point B receives light from all parts of the source except for the blocked bundle AP. The intensity distribution on the object plane is Iobject(x,y) = Jo

— ff

J(.f, g) d.fdg,

where S2 is the solid angle that P subtends at (x, y), and f and g define the propagation direction of the light beam AP. For a small opaque spot, Iobject(x,y)

= IO — C J(f, g)

with c being a constant. The relationship between (x, y) and (f, g) is (see Exercise 9.1): nob'ectsin 15

nglass r

_(9.8)

t 2 + r2

where r = x2 +y2 , sine = \/ f2 + g2/f, and t is the thickness of the reticle. If we define Jmax as the maximum intensity of the effective source and Ia n as the corresponding minimum intensity on the object plane, then Irvin = Jo — cJ , and f(f,g) Jmax

Io I(x,y) — lo — 'min —

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Pupil



179

object plane

Figure 9.13: An opaque spot on the back of a clear reticle creates a reverse-tone image of the effective source on the image plane.

The intensity distribution of the effective source is replicated in the object, which is in turn reproduced onto the image plane.

9.6 Pupil Aberration contributes to fluctuation in two ways: image variability caused by aberration itself, and fluctuation resulting from across-field aberration changes. In the presence of coma, for example, the images of a pair of nominally identical spaces described by if d/2 < Ill < 3d/2 tfg (9.9) Óx(z) _ tb g otherwise, are different. This asymmetry changes as the amount of coma varies across the image field.'° Vignetting is another defect that can cause fluctuation. Usually the result of cursory lens design, some image-forming light rays are not captured by the lens when vignetting occurs. The peripheral of the field is especially prone to this defect.

9.7 Focus Focus variation arises from focal plane deviation (variation of Z4 across the image field), astigmatism fluctuation (variation of Z5 and Z6 across the exposure field), wafer topography, mask flatness variation, focus setting and auto-leveling errors, wafer and chuck nonflatness, lens heating, barometric pressure and other environ1 °Brunner [100], Progler and Wheeler [101], and Flagello, et al. [102] provide more discussions on the lithography effects of aberrations.

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Variabilities

180

mental variations. A defocused image is blurred, resulting in image quality degradation. Although decreasing X and increasing NA improve resolution of imaging systems, these measures adversely impact depth of focus—the maximum amount of tolerable focus variation. We can estimate how depth of focus scales with 4 and NA using Rayleigh's criterion [103]. Consider the situation shown in Fig. 9.14, where two light rays, one from the center of the pupil and the other from the edge of the aperture, interfere to form a sharp image on the plane zo. This image degrades as the observation plane moves away from zo. At a plane a distance z from zo, the relative phase change between the two rays is 21t

nimage (z — z cos 0obj)

rad.

Rayleigh used the heuristic criterion that the image is sufficiently blurred when the phase change is 90 deg, namely, the optical path difference is a quarter wavelength. According to this criterion, the Rayleigh unit of depth of focus is a distance R. U. such that (see Exercise 9.2) R. U. = -0 1

2 I

1 + 1— sin e 8obi /

4 nimage sin B obt \ 1 1

4 NA • sin S abi

(9.10) 1 — sin2 9abj .

In improving the resolution by reducing X and increasing NA the focus tolerance diminishes. The decrease is linear with wavelength, but depends on the manner in which the numerical aperture changes. Increasing the image space refractive index decreases depth of focus linearly with 1/ni mage , namely, linearly with 1/NA. On the other hand, increasing the semi-aperture angle reduces focus tolerance faster than 1 / sin 2 9 obj, namely, quadratically with 1/NA. When the semi-aperture angle is small, the Rayleigh unit is approximately R. U. =

2 sint 0obj

(9.11)

As sin0 obi approaches one, the Rayleigh unit approaches (X/4) rather than (?/2). The depth of focus is further halved because of the (1 + 1— sin e A obi) factor. Obtaining a larger numerical aperture by increasing nimge impacts depth-of-focus less adversely than increasing sine o b^. °Although defocus is one form of aberrations, we consider it separately because of the varied causes and its historical importance in optical lithography.

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Focus



181

pupil ^,' 1 0 1 — 0 2 1= ^/4

eob; nimage

I 01 1 z o z-zo nimage

Figure 9.14: Rayleigh's depth of focus criterion.

Figure 9.15: The focus monitor comprises lines bordered by transmitting regions

that are 90deg out of phase. Similar to the definition of kl for minimum image size [see Eq. (5.27)], we use the k2 factor to denote the depth of focus of a lithography process [104]: depth of focus = k2 R. U. k2

2

for low-NA imaging. 2 sin 9 obj

Focus sensitivity of a process and hence its difficulty increase with decreasing k2. We can measure focus crudely by exposing isolated lines of decreasing dimensions onto a positive photoresist through a series of focus levels [105]. The focus value that prints the narrowest line is the best focus. For finer determination of focus, we can use the focus monitor [106]. The measurement structure consists of a line bordered by two transmitting regions that are 90 deg out of phase, as shown in Fig. 9.15. The 90 deg phase makes the lines Ll and L2 shift in opposite directions with focus changes (see §9.3.2), and the displacement, being insensitive to dose fluctuation, is linear with respect to defocus at small deviations from the diffraction focus. Together with the lines L3 and L4 this structure can be used to characterize focus and astigmatism.

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182



Variabilities

á^ f°o E m

E

x (7/NA)

- f (NA/A)

(a) transmissivity

x (a/NA)

(b) spectrum

(c) approximation

Figure 9.16: (a) A transmission wedge for monitoring dose variation. (b) Its spectrum. (c) Its approximation.

9.8 Dose Dose change can cause both inherent variation and fluctuation. On the one hand, dose nonuniformity arising from wafer topography, wafer stack, and exposure dose variation, including actual dose fluctuation across exposure field or across the slit of step-and-scan exposure systems, and effective dose variation caused by nonuniform hot plate temperature during post-exposure bake, results in fluctuation because of finite exposure latitude. On the other hand, a uniform dose change such as a deliberate dose modification or a dose setting error affects proximity effect because dose sensitivity (exposure latitude) differs among features. The ideal structure for dose monitoring should be sensitive to dose variation but not to other process detractors. One candidate is a transmission wedge described by Ôx(x) — m(1—mpzI)

0

if ^zI < m,m>0, (9.12) otherwise,

with the spectrum (see Exercise 9.4) m2 [1—cos(27tf/m)]. Ó,f (f) = 22f2

(9.13)

The transmission function and spectrum of an example structure with m = 0.2 is shown in Fig. 9.16(a) and (b). Because of the inverse f^ dependence the spectrum is sizable only around f = 0. Concentration of energy in low-spatial-frequency components implies low sensitivity to aberrations including defocusing. But the gradual transmissivity change ensures responsiveness to dose variations.

The transmission function of Eq. (9.12) is difficult to realize in practice. We can approximate the gradation by a structure comprising an array of spaces at a fixed period but decreasing duty ratio from the middle, as illustrated in Fig. 9.16(c) [1071. This pattern is sensitive to actual and effective dose variations including maskdimensional error.

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Flare

183

We can determine dose and focus concurrently using the two structures shown in Fig. 9.17 [108]. One of the patterns, shown in Fig. 9.17(a), consists of two gratings situated in a clear background. The grating period p is beyond resolution by the optical metrology system, namely, p

metrology o < 2NAmetrology

The image varies only in the horizontal direction [along AB in Fig. 9.17(a)], as plotted in Fig. 9.17(c); and we can associate with this structure a width w s . An analogous quantity wl, corresponding to two gratings situated in an opaque background [Fig. 9.17(b)], can be similarly determined [Fig. 9.17(d)]. With increasing defocus, the grating features of both structures shorten, causing ws and w1 to increase. Their focus dependence is approximately quadratic around optimum focus, as shown in Fig. 9.17(e). With dose increase, however, the grating features of Fig. 9.17(b) lengthen linearly (on positive photoresists) while those of Fig. 9.17(a) shorten, causing the behavior illustrated in Fig. 9.17(f). Based on the similarity in focus response and difference in dose behavior, we can develop a parametric model such as the following [109]: wti = ao ; + ai 1 E + (a21 + a3 ; E) (z — zo), i E {l, s}, where the coefficients ao 1 ,...,3, and the best focus position zo are parameters determined by experimental calibration. With these parameters, the focus [or defocus (z — zo)] and dose E can be solved for each pair of measurements w s and wl:

E_

—b+ b 2 — 4ac

2a

and

(z — zo) 2 =

wi

— (ao i +a1 E) ;

a2, +a3 E ;

where

a = al,a3, — a3 1 a1, b = a3 1 WS — a3 wt +ao 1 a3 +al 1 a2, — a2 1 al s — a3,ao., c = a2 1 ws —a2swt +ao l a2, —a2 1 aos . S

5

9.9 Flare Also called stray light and scattered light, flare is light existing in the image beyond that captured in the physics described by Eq. (8.1). Figure 9.18 illustrates the primary sources of stray light. Consider image formation of an object point A with the geometrical image point B. Because of diffraction and possible focus error, the intensity distribution around B is blurred. A fraction of this halo is reflected from the wafer toward the reticle and then back to the wafer from the reticle, adding to the original image intensity. Each additional round-trip reflection increases the amount

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184

Variabilities

Figure 9.17: The similarity in focus response (e) and difference in dose behavior (f) between the structures shown in (a) and (b) enables simultaneous determination of focus and dose.

of stray light, with the halo increasing in size but decreasing in brightness. Depicted by the dotted rays in Fig. 9.18, we can reduce this cause of flare by decreasing the reflectivity of the wafer or the reticle or both. Light scattering from optical elements also contribute to flare. Scattering is caused by surface roughness of optical elements. An ideal optical element has smooth surfaces; it performs its designed function without scattering. For example, a thin lens with smooth surfaces focuses a parallel light beam toward a point on the focal plane within the approximations of geometrical optics (see §2.4). There may be undesired specular reflection if the lens anti-reflection coatings are imperfect;

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Flare



185

wafer and reticle

-

reflection

B A ^ ^ y

, -

------------surface

D

- aberration JC

scattering

Figure 9.18: Primary sources of flare.

but no light is diffused into other directions. However, a physical optical element has (microscopically) rough surfaces that scatter light. The amount of scattering depends on surface roughness and wavelength according to e(4na/X)2 — 1

where a is the standard deviation of the assumed Gaussian-rough surface [110]. Any light (including image-forming beams and rays bouncing between the wafer and the reticle) that impinges onto a surface is scattered; and the amount of scattered light increases with the number of optical elements. Stray light also arises from high-order aberrations. Recall the discussion in §7.4 that the ray aberration generally increases with the Zernike term number, causing the diffraction pattern to spread. If a wavefront contains significant components of high-order aberrations that are not included in the truncated Zernike-series representation of Eq. (7.17), the diverted light contributes to flare. This source is depicted by the light ray ACD in Fig. 9.18. We can surmise that stray light generally increases with the average transmission of the object, and the amount of flare at a particular field point depends on the magnitudes of these sources and their characteristic light spreading distances [111]. Scattered light has the longest range (on the order of the field size), while ray aberration has the shortest (on the order of micrometers). The amount of flare is a function of object pattern density over a large area. By raising the overall image intensity, stray light reduces contrast. But flare variation across the field causes more fluctuation than contrast loss. As an example, assume a feature prints at a nominal dose of Dprint = 1OD0/3, where Do is the dose to clear a large area of positive photoresist. The corresponding normalized intensity level (threshold intensity) is 3/10 = 0.3. If the flare decreases by 3%, the feature would be printing at a normalized intensity level of 0.30 + 0.03 = 0.33 of the original image. This 3% flare level change translates to a 10% dose variation; Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms

Variabilities

186

the effect of flare variation is amplified. Let us call this effective magnification the flare-amplification factor (FAF): FAF = D^ón

t

Given by the ratio of dose-to-print to dose-to-clear, the flare-amplification factor causes large amounts of fluctuation especially for features with little exposure latitude. By direct addition of light to images, flare causes features to print at a different intensity level, resulting in gross fluctuation. A simple way to measure flare is to expose an opaque object, so large that its nominal interior intensity is zero in the absence of flare, in a sequence of dose steps and observe the dose Ddisappear at which its image formed in positive photoresist completely disappears [Ill]. The ratio of dose to clear D o to Ddisappear is the amount of flare: flare = D0 x 100%. (9.14) Ddisappear

A more detailed characterization of flare can be accomplished by using a large opaque pattern with small openings placed at regular intervals [111], as shown in Fig. 9.19(a). We again expose this test pattern in a sequence of dose steps onto a positive photoresist. As dose increases the edge of the developed photoresist recedes from the designed edge. From the dependence of the resist edge position, determined relative to the small openings, on the exposure dose, we can deduce, using an equation similar to Eq. (9.14), the amount of flare as a function of distance into an opaque region. Figure 9.19(b) shows the result of such a measurement. Three distinct regions are observed. At large distances, the amount of stray light steadies around 3.2%; the background flare (or long-range flare) is 3.2%. At intermediate distances between 1µm and a few tens of micrometers, flare decreases gradually with distance. Sometimes called midrange flare, this behavior is attributable to high-order aberrations [ 112] and wafer-reticle reflections. At distances smaller than 1µm the change in spatial characteristics indicates flare arising from different phenomena. We cannot draw definite conclusions, however, because accuracy of this receding edge method worsens with increasing proximity between the photoresist and geometrical edges due to diffraction and lateral photoresist development effects.

9.10 Remarks We can simulate, with a program such as one developed using the equations of Chapter 8, the effects of most causes of variabilities. Their impact is naturally object-dependent. Even for a specific technology for which only one critical dimension is of primary concern (namely, a single kl factor), image sensitivity can vary greatly among different pattern configurations. Nevertheless, it is possible to

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187

Remarks

0

CO

0 large opaque pattern

\

mid-range

0 00

CO

O Q0

long-range

0000

V small openings

short-range

Co 4

0

N -

0.1

1.0

10.0

100.0

distance from designed edge (gm) (a) test pattern

(b) measurement results

Figure 9.19: (a) Test pattern for the receding edge method. (b) An example of mea-

surement results [111 ]. estimate a representative sensitivity for each detractor and subsequently a corresponding control requirement. An exercise performed for a kl = 0.35 technology gives the following estimations: Detractor Proximity effect Mask dimensional error Phase error Transmission error Mask edge roughness Degree of polarization Non-telecentricity Aberrations Focus Dose Flare

Control Requirements OPC ±0.008/M (?o/NA) ±2deg 0.005 (Do)

— < 0.05 (for unpolarized illumination) ±0.01 (p) 0.015 (2 rms) ±0.5 (R. U.) [see Eq. (9.10)] ±2% (Dprint) ±0.5% (Do)

These requirements are determined separately for each variability contributor. During actual processing all detractors affect imaging and their net effects are quantified by metrics such as across-chip linewidth variation (ACLU). The manner in which variabilities from various causes combine is a question that often arises in tasks such as estimating variability, developing control requirements, and deciding how the different components of each cause mix together. (An example of the latter task is the determination of how focal plane deviation, wafer topography, mask flatness variation, focus setting error, and wafer nonflatness combine into an overall focus error.) In one extreme we can assume that variabilities caused by all

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Variabilities

188

contributors add linearly. But this conservative and unrealistic assumption results in unnecessarily stringent control requirements. The other extreme assumes that the effects are uncorrelated and random, such that variance of the total across-chip linewidth variation is the sum of the individual detractor variances: t

ótotal =

6i

je {detractors}

This latter assumption is usually adopted in practice. We should beware, however, that effects of independent causes may unwittingly reinforce or cancel one another such that the uncorrelation assumption is violated. For example, the spatial linewidth variation signatures resulting from mask dimensional error, aberrations, and flare fluctuation can have much semblance [113]. In general, individual variabilities and their possible correlation is process dependent. They can be determined only by thorough empirical characterization [113-115]. A well-characterized process would also allow us to perform optical and process correction (OPC), i the technique of predistorting mask patterns to reduce image variability [116-125]. Being a photomask approach, OPC can only remedy inherent variations and systematic and stable fluctuations. We should minimize random variability components for efficacious correction. Even with exact engineering control, however, noise is one random fluctuation that cannot be eliminated. In the context of optical lithography, shot noise refers to the statistical variation in the number of photons used to define a feature. The impact of shot noise depends on the photoresist, the photoacid generation mechanism, and the post-exposure bake and dissolution processes [126-128]. To understand qualitatively the effects of shot noise, consider 193 -nm exposure of a contact hole of area A = 45 nm x 45 nm with a dose of Dp int = 1 mJ cm 2 . The number of photons absorbed Nphotons is approximately h

Nphotons

Dprint X Ithreshold X fabsorbed

xA

Ephoton

where fabsorbed is the fraction of exposure energy deposited into the photoresist, Ithreshold is the threshold intensity of the contact exposure process, and Ephoton is the energy carried by each photon. Assuming the resist absorbs 10% of the exposure energy and Ithreshold = 0.3, 10-3•0.3•0.1•2025 x 10 Nphotons =

-

14

x 10 +$

6.6256 X 10-34. 3193x10- 9 590.

There are only 590 = 24 photons per side, with each corresponding to a substantial 4.1% of the critical dimension. Stochastic variation in the number of photons `''OPC was originally shorthand for optical proximity correction. But the correction has since been applied to compensate for nonoptical processing effects.

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Exercises



189

absorbed can cause resist line-edge roughness, resulting in higher variability. Such line-edge roughness can be reduced by increasing the number of absorbed photons, thereupon lowering the photoresist speed.

Exercises 9.1 Derive Eq. (9.8). 9.2 Derive Eq. (9.10). 9.3 In the low-NA limit, what is the Zernike series representation of one Rayleigh unit? 9.4 Derive Eq. (9.13).

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Appendix A

Birefringence The theories and equations explicated in the text were based on two distinct sets of fundamental equations: Maxwell's equations [Eqs. (1.1)—(1.4)], which governs wave propagation; and constitutive relations [Eqs. (1.7)—(1.9)], which describe the interaction of electromagnetic waves with materials. We have assumed in our developments that materials are optically isotropic such that the electric displacement D and electric field E are related by the dielectric constant £e , and the magnetic field H and magnetic induction B are related by the permeability µ,n : D = £ e (w)E

and

Bim(CO)H.

(A.1)

The permittivity and permeability are frequency dependent; however, they can be considered constants since we are concerned with monochromatic waves. The constitutive relations described by Eq. (A. 1) indicate that the electric field is aligned with the displacement, and the magnetic field with the induction. In an anisotropic material, however, these pairs of vectors are not parallel. For electrically anisotropic but magnetically isotropic materials, we can retain the relationship between the magnetic field and induction in Eq. (A. 1), but the simplest connection between the electric field and displacement becomes a linear matrix transformation: D=

Dx

[EJ

£ems £e Xy E e l eyx £ eyy LeyZ Ey

= I£

£ezy £e z ,

= £e E,

EZ

where the nine quantities £ejj constitute the dielectric tensor £e .' Let us investigate consequences of misalignment between the electric field and displacement on wave propagation. For a monochromatic plane wave of angular frequency co, the field phasors are proportional to exp(ice[ --r.s] —t). 'Born and Wolf [129] contains detailed discussions on optical behavior in anisotropic materials.

191

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192



Birefringence

B H

S

E

E1

'

D Figure A.1: The electric field E and electric displacement D are not parallel in an electrically anisotropic medium.

We can express the partial derivatives of the field vectors as

á

E = —iwwE,

and

all

= — iwH,

V x E = tw—ss x E,

VxH=iwnsxH.

In the absence of currents, Eqs. (1.1) and (1.2) become H=— n 9xE

and

JLmCO

D= n9xH. co .

(A.2)

Substituting the first expression of Eq. (A.2) into the second and using the vector identity Ax(BxC)- (A.C)B—(A•B)C gives n2

n2

n2

m C0

mC0

UmC0

D =— 2 sx(s"xE)= 2 [E—(9 -E)s]= 2 E 1 .

( A.3)

The quantity E1 denotes the vector component of E perpendicular to in the plane formed by E and s, as illustrated in Fig. A. 1. Since, from Eq. (A.2), the vectors H and B are perpendicular to E, D, and s, the latter three vectors are coplanar, with D being orthogonal to "s. The magnetic field H and the electric displacement D are transverse to the normal of the wavefront 9, as in an isotropic medium; but the electric field E is not. To understand the manner in which wave propagates in anisotropic materials, we revisit the expressions for the electric and magnetic energy densities [Eq. (1.14)]

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Birefringence

193

and the Poynting vector [Eq. (1.28)]: 1 2

1 we = 1 E • D = 1 Ei£eijEj,

2i

(A.4)

={x,y,z} j={x,y,z}

w m = ZH•B= 2 IH^ 2 ,

and

S=ExH. Performing mathematical manipulation similar to that leading to Eq. (1.12), we derive that —V.(ExH)=E.-D+H aB at at =

aEj u„

óM1 2

Ei£e `^ 2 + 2 at i ={x,y,z} j={x,y,z}

The term on the left side is the rate of total energy density increase. In the absence of absorption or amplification, it should be the sum of the rate of electric and magnetic energy density changes. For the first term to represent the rate of electric energy density change, it must equal the time derivative of Eq. (A.4): aEj1

aE j aEi

[^

1 Ei£e at = 2 L £e"( Ei —+E —) at at ° i ={x,y,z} j={x,y,z} i ={x,y,z} j={x,y,z}^' 1

meaning 1 aEi

aE j

1

at

i ={x,y,z} j={x,y,z}

£e`' (E —

El

at

=

^+

(EejjEejj)Ei

L

0.

i ={x,y,z} j={x,y,z}

For nonzero electric field components, the above expression is zero if, and only if, £e jj = £ ejj .

The dielectric tensor is symmetric. It has six instead of nine independent components. We can now write the electric energy density as

2

£ E Z £e E? £ E?

We = e 2 x +

Y

+

e

2

+ £eyz EyEZ + £exz ExE2 + Eeg ExE9.

(A.5)

Since the energy density must be greater than or equal to zero, Eq. (A.5) is a semipositive-definitive quadratic form. Spatial points with the same energy density form

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Birefringence

194

an ellipsoid surface. There exists a coordinate system (x',y',z') such that the energy density is We = Eex,Ex +£ey Ey, + Eez E^, where C ex„ E ey„ and Eeg, are the principal dielectric constants (or principal permittivities). In this system of principal dielectric axes, the material equations are simplified: D'

=

E ez, ExI ,

Dyi = Eey, Ey ,

and

DZ' = Cez, Ezi .

(A.6)

Because the principal dielectric constants are distinct for an anisotropic material, the electric field E and displacement D are not parallel unless E coincides with one of the principal axes." Substituting Eq (A.6) into Eq. (A.3) gives, after some straightforward algebra, skEk

n 2 ( g

E )sk2

, — n2 —^um£ekCO

k E {x, y', z' }.

(A.7)

Adding this set of three equations together and dividing the sum by (s • E) results in k={ ',z'}

Subtracting both sides by

n2s2 k um6ekCO

n2_

sk = 1, we obtain k={x' y',z'}

2 2

2 2

(A.8)

f ____OSk v = 2 psk 2 2 2 k={x',Y ,z'} n —,Um£ek CO k={x',y',z'} vk — Vp

where V P = co/n and vx, _

1

vy1 _

1

um£e^

and

vz,

1 ,umEeZ,

are the three principal propagation velocities. Equation (A.8) is a quadratic equation in vp. There are two values of v p for every direction s". With each of the two values of v p , we can solve Eq. (A.7) for the ratios between the electric field components, and subsequently find the electric displacement components using Eq. (A.6). Since the ratios between the field components are real, the electric field and displacement are linearly polarized. We can conclude that an anisotropic medium permits two monochromatic plane waves with two different linear polarizations propagating with two different velocities in any given direction. The electric field also aligns with the displacement if E e,, _ £ey, = se . But this is the isotropic case.

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195

Birefringence

A ramification of having two propagation velocities is a phenomenon known as double refraction or birefringence. When a plane wave incidents from an isotropic medium onto an anisotropic one, there are generally two refracted rays on the plane of incidence, since, according to Snell's law [Eq. (6.22)],

e

Si n int

_ Cinc

sin 8tran Ctran ctr and hence 6tran can assume two values.' In photolithography lenses made of birefringent materials (such as calcium fluoride for 157-nm lithography), each light ray entering an imaging system with N lens elements gives rise to 2N beams in the exit pupil. Such birefringence can be described by a 2 x 2 matrix that characterizes coupling between the different waves [130, 131 ].

"'In addition to intrinsically anisotropic materials, birefringence can be caused by anisotropy induced in nominally isotropic media resulting from mechanical stress, or by an orderly arrangement of particles of isotropic materials. The former phenomenon is called stress birefringence, while the latter is called form birefringence.

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Appendix B

Stationarity and Ergodicity l In computing time averages of optical fields, we regard the ensembles to be stationary and ergodic. Stationarity means that all ensemble averages are independent of the time origin; and ergodicity implies that each ensemble average is equal to the time average involving a typical member of the ensemble. If we denote the field at a point Pi arising from a point source i by Ul (Pi, t), < U(P1,t) >= f Uip(Ui)dUi = 0,

since Ul (Pi, t) can be viewed as a random process on a time scale much longer than the coherence time of the light. Now consider another field generated by a different point source j. Let us denote the field at P2 arising from j by U1 (P2, t). The time average of U, (Pl , t) U1 (P2, t) is +00

< Ui(P1,t)Uu(P2,t) >= ffuuJp(u,uJ)duduj.

Since U (Pl , t) and U^ (Pa, t) are statistically independent, P(U, U.i) = P(Ui)P(U1),

and < Ui(Pi,t)Uj(P2 i t) > = fUp(U)dUiJUjp(Uj)dU1 =< U(P,,t) >< U;(P2 i t) > =0. 'This appendix is contributed by Anthony Yen. 197

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Appendix C

Some Zernike Polynomials j

n

m

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

0 1 1 2 2 2 3 3 3 3 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8

0 1 1 0 2 2 1 1 3 3 0 2 2 4 4 1 1 3 3 5 5 0 2 2 4 4 6 6 1 1 3 3 5 5 7 7 0

an

R,`, (0)YJ (0)

vÏ \ V6 \

v

V 10 10 10

viö 12 12 12 12 VÏ2 12 14 14 14 14 14 16 16 16 16 16 16 VÎ 6 16

(+1) (+p)cosrp (+p)sin4 (+2 2 _1) (+p 2 )sin24 (+p 2 )cos24 (+3p 3 -25)sin4 (+3p3 — 2p)cos4 (+p3)sin34 (+53)cos34 (+6p 4 -65 2 +1) (+40 4 - 3p 2 )cos2¢ (+40 4 — 3p 2 )sin24 (+p 4 )cos44 (+p 4 )sin44 (+10p 5 —12pá +3p)cos4 (+100 5 —120 3 +3p)sin4 (+50 5 -4p 3 )cos3c (+50 5 —4p 3 )sin30 (+p 5 )cos54 (+p 5 )sin54 (+2006 — 3004 + 120 2 — 1) (+150 —200 4 +6p 2 )sin20 (+150 6 — 200 4 + 6p 2 )cos 20 (+60 6 - 5p 4 )sin44 (+60 6 — 5p 4 )cos 44 (+p6)sin64 (+p )cos64 (+350 7 — 600 5 + 305 3 — 4p)sin 4 (+350 7 —600 5 +300 3 —4p)cos4 (+210 7 — 300 5 + 10p 3 )sin34 (+210 7 — 300 5 + 100 3 )cos 34 (+70 7 - 615 5 )sin50 (+70 7 - 60 5 )cos54 (+p 7 )sin7m (+p 7 )cos74 (+7008-14006+9004-2002+1)

199

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Some Zernike Polynomials

200

JÎ8

38 39 40 41 42 43 44 45 46 47 48 49 50

8 8 8 8 8 8 8 8 9 9 9 9 9

2 2 4 4 6 6 8 8 1 1 3 3 5

51

9

5

52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93

9 9 9 9 10 10 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 12 12 12 12 12 13 13

7 7 9 9 0 2 2 4 4 6 6 8 8 10 10 1 1 3 3 5 5 7 7 9 9 11 11 0 2 2 4 4 6 6 8 8

10 10 12 12

1 1

18 iT8 18 VÎ 8 18 18 18 20 20 20 20 20 20 20 20 20 20 11 22 22 22 22 22 22 22 22 22 22 24 24 24 24 24 24 24 24 24 24 Vi 24 VT3 26 v26 26 26 26 26 26 26 26 26 26 26 28 28

(+560 8 —10515 6 +600 4 —10p 2 )cos24 (+5615 8 —1050 6 +6004 —10p 2 )sin 2cß (+2815 8 - 420 6 +1515 4 )cos40 (+280 8 — 420 6 + 15p 4 )sin44 (+80 8 - 715 6 )cos6c (+80 8 —7p 6 )sin60 (+15 8 )cos80 (+p 8 )sin84 (+1260 9 -2800 7 +2100 5 - 600 3 +5p)cos4 (+1260 9 - 2800 7 +2100 5 - 600 3 +5p)sin0 (+840 9 —1680 7 + 1050 5 — 200 3 )cos 34 (+840 9 —1680 7 + 1050 5 — 200 3 ) sin 30 (+360 9 - 560 7 +2115 5 )cos54 (+360 9 - 560 7 +2115 5 )sin5 0 (+90 9 — 8p 7 )cos74 (+90 9 -80 7 )sin70 (+15 9 )cos90 (+15 9 )sin90 (+2520 10 — 6300 8 + 5600 6 — 2100 4 + 300 2 —1) (+2100 10 —5040 8 +4200 6 —1400 4 + 15p 2 )sin24 (+2100 10 — 50415 8 +4200 6 —1400 4 + 15p 2 )cos 20 (+1200 10 - 2520 8 +1680 6 - 35p 4 )sin44 (+ 1200 10 — 25215 8 + 1680 6 — 350 4 )cos 40 (+450 10 - 720 8 +28p 6 )sin60 (+45 0 10 — 720 8 + 28p 6 )cos 60 (+10p 10 - 90 8 )sin80 (+10p 10 - 90 8 )cos80 (+p 10 )sinl04 (+15 10 )cos 100 (+4620 11 12600 9 + 126015 7 — 5600 5 + 1051 3 — 615) sin 0 (+4620 11 -12600 9 + 12600 7 — 5600 5 + 1050 3 — 6p)cos 4 (+3300 11 — 8401 9 + 75615 7 — 2800 5 + 350 3 ) sin 34 (+3300 11 -84015 9 +75615 7 -28015 5 +35p 3 )cos30 (+1650 11 — 3600 9 + 2520 7 — 560 5 ) sin 50 (+1650 11 3600 9 +2520 7 - 56p 5 )cos51 (+55p 11 — 900 9 + 360 7 ) sin 70 (+550 11 — 900 9 + 36 5 7 ) cos 70 (+110 11 100 9 )sin90 (+110 11 1015 9 )cos94 +p l l )sin 114 (+P 11 )cos 110 (+9240 12 — 27720 10 + 31500 8 —168015 6 +42015 4 _4215 2 +1) -

-

-

-

(

(+7920 12 — 23100 10 + 25200 8 —12600 6 +2805 — 210 2 )cos 20 (+7920 12 — 23100 10 + 25200 8 —12600 6 + 2800 4 — 21 15 2 )sin 21 (+49515 12 -13200 10 +12600 8 -5040 6 +7015 4 )cos4 0 (+4950 12 —132015 10 + 126015 8 — 5040 6 + 700 4 ) sin 40 (+2200 12 — 4950 10 + 3600$ — 840 6 )cos 60 (+2200 12 - 4950 1 0 +360p 8 - 8415 6 )s1n64 (+660 12 -110p 10 + 45p 8 )cos80 (+660 12 -110p 10 + 450 8 )sin80 (+120 12 —11p 10 )cos 100 (+ 120 12 —11p 10 )sin 100 (+15 12 )cos 120 (+0 12 )sin 120 + 12600 5 —1680 3 + 7p)cos 0 (+17160 13 — 55440 11 + 69300 9 —420015+ (+17160 13 — 554415 11 + 69300 9 — 42000 7 + 12600 5 —16ßp 3 + 715) sin 0

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Some Zernike Polynomials

94 13 3 95 13 3 96 13 5 97 13 5 98 13 7 99 13 7 100 13 9 101 13 9 102 13 11 103 13 11 104 13 13 105 13 13 106 14 0 107 14 2 108 14 2 109 14 4 110 14 4 111 14 6 112 14 6 113 14 8 114 14 8 115 14 10 116 14 10 117 14 12 118 14 12 119 14 14 120 14 14 121 15 1 122 15 1 123 15 3 124 15 3 125 15 5 126 15 5 127 15 7 128 15 7 129 15 9 130 15 9 131 15 11 132 15 11 133 15 13 134 15 13 135 15 15 136 15 15



201

28 (+12870 1 3 -39600"+46200 9 -25200 7 +630p á -560 3 )cos30 28 (+12870 13 —3960" +46200 9 —25200 7 +6300 5 — 563 3 )sin34 28 (+7150 13 —19800" + 19800 9 — 8400 7 + 1260 5 )cos 50 28 (+71áp 1 ' — 19800" + 19800 9 — 8400 7 + 126p 5 )sin50 28 (+2860 1 ^ — 6600" +4950 9 —1200 7 )cos 70 28 (+2860 13 -6600 11 +4950 9 -120p 7 )sin70 28 (+780" 1 3 -1325 11 +55p 9 )cos94 28 (+780 13 —1320" +55p 9 )sin94 28 (+130" 13 —120")cos 110 28 (+130 1 3 — 120 11 )sin 114 28 (+p' 3 )cos 134 28 (+p 13 )sin 130 vTi5 (+34320 14 -120120 12 +166320 10 -115500 8 +42000 6 -7560 4 +560 2 -1) 30 (+30030 14 — 1029615 12 + 138600 10 — 92400 8 + 31500 6 —50415 —5040 4 +28 0 2 ) sin 20 30 (+30030 34 — 102960 12 + 1386015 10 — 92400 8 + 31500 6 — 5ß4p 4 + 28p z )cos 24 30 (+20020 14 — 64350 12 +792015 10 — 46200 8 + 12600 6 — 12615 4 )sin40 30 (+20020 14 -64350 12 +79205 10 -46200 8 +12600 6 -126p 4 )cos441 30 (+ 100115 14 -28600' 2 +29700 10 -13200 8 +2101 )sin60 30 (+10010 14 -28600' 2 +297015 10 -13200 8 +21015 6 )cos641 30 (+3640 14 — 8580 12 + 66015 10 —1650 8 )sin 80 30 (+36415 14 — 8580 12 + 6600 10 —16515 8 )cos 8i 30 (+910 14 -1560 12 +660 10 )sin100 30 (+910 14 —1560' 2 +66p 10 )cos 100 30 (+140 14 -1315' 2 )sin120 30 (+140 14 -1315' 2 )c0s120 30 (+p 14 )sin140 30 (+15 14 )cos140 32 (+64350 15 _2402415 13 +3603615h 1 — 277201 9 + 1155007 —2520i5 +25215 — 8p)sin 0 32 (+64350 15 — 240240 13 +360361511 —2772015 0 + 1155015v —252015 +25215 — 815)cos o 32 (+500515 15 — 180180 13 +25740151 —184800 9 +693015 v — 12601 5 + 8415 3 )sin 341 32 (+50050 15 — 180180 13 + 257400 11 — 184800 9 + 69300 7 - 12600 5 + 840 3 )cos 30 32 (+30030 15 —100100 13 + 128700 11 — 79200 9 +231015 7 — 25215 5 )sin50 32 (+30ß3p 15 — 100100 13 + 128700 11 —792015 +23100 7 — 252p 5 )cos50 32 (+13650 15 -40040' 3 +429ßp"-19800 9 +33013 7 )sin74 32 (+136515 15 -40040 13 +42900 11 -19800 9 +330p 7 )cos7, 32 (+45515 15 -10920 13 +8580"-22015 9 )sin94 32 (+45515 15 — 109215 13 + 8580 11 — 2200 9 )cos 90 32 (+10515' 5 -1820 13 +7815")sin110 32 (+ 1050' 5 —1825 13 +780")cos 110 32 (+150 1 5 —1415 1 3 )sin 130 32 (+ 1515 15 — 14p 13 )cos 134 32 (+15 15 )sin 150 32 (+15 15 )cos 154

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202

Some Zernike Polynomials

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Some Zernike Polynomials

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203

I

Appendix D

Simulator Accuracy Tests It is often desirable to assess the accuracy of an imaging simulation program. A test suite for fidelity evaluation should be comprehensive, and the cases should admit analytic solutions. In this chapter we construct a set of tests, based on the physics encapsulated in Eq. (8.1), which can be used to gauge the accuracy of simulation programs.

D.1 Blank mask For a mask that is completely transmitting,

Ô(1,9)=1

6 (f,) = S(Í)S(8)

and

Because of the obliquity factor (see §4.2), the coupling image intensity, although constant, is not necessarily unity: +00

I^1,9,z) —

2 1

ff

—

J(f,)

nimM age M2"2e

_ P e

dfd8•

For radially symmetric illumination described by i 1(1,g)

if Ginner P C bouter,

100'ute

0

otherwise,

the intensity is

_

bouter Sln eobj

2^ S

— 7C Sin t bob ( outer uter

=

6inner 2 )

J

1 — n i2tnageM2I5 1— p2

PedPe

Ginner Slrieobj

1-6 ener Slri z eobi

nimageM

1 + a dx,

sin t Bobj (óouter — 6 inner)

i — bouter Sin t eobj

x

205

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206



Simulator Accuracy Tests

where x = 1-

pé and a = 1 / ( n

ge M 2 )

- 1. The indefinite integral is analytic:

1 + a dx = z+ a ln(a/2+x+z), with z = x(a+x). The intensity is

2

I = 2 ai geM sin eobj (bouter -

Z+ a ln(a/2+x+z) ó inner) 2

1-6 ener s in t 9 obj 1-6o„ters1n2Oobj

Table D. 1 tabulates the intensities of various imaging systems. Table D.1: Coupling image intensities of a 100% transmitting mask imaged by various systems.

nimage

1

NAóinner

NAóouter

sin9 o bi 0.0

sinO o bj 0.0

1

1

1

0.2 0.4 0.6 0.8 0.4 0.6 0.8 0.6 0.8 0.8

1 1 1 1 1 1 1 1 1 1

1.0095713537 1.0409095183 1.1046046472 1.2363625785 1.0513555731 1.1164838089 1.2514819934 1.1555607504 1.3015135985 1.4057656329

1.0097995793 1.0418649964 1.1069527619 1.2412943339 1.0525534688 1.1190969098 1.2567273175 1.1590229744 1.3077707797 1.4140192121

0.0 0.2 0.4 0.6 0.8 0.4 0.6 0.8

1 0.9870433935 0.9415567288 0.8274118142

1 1.0087783694 1.0375828099 1.0963999155 1.2190366429 1.0471842901 1.1073526087 1.2330538611

1 1.0092923179 1.0397401291 1.1017257759 1.2303000474 1.0498893995 1.1132799582 1.2450338960

0.2

0.4 0.6 1.5

0.0

0.2

M=1

0.9263945072 0.8074578668

-

M

= 0.25

M

= 0.2

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Images of M = 1 systems

207

D.2 Images of M = 1 systems For aberration-free unity-magnification systems in which the object and imagespace media are the same, the scalar-coupling image is given by a simplified form of Eq. (8.3): —i2 Î(1,.9,z) = I I Cm' W WOmn n, e

TCCm' n'; mx nrf,

n",m" n',m'

where / mr nr m` 0 l --- 1^circ —,-,1 dfdg, Py ( P.„ Icy +_

TCCmn,m"n" =ff f(J)circ

px

(D.1)

and circ(fo go 6) = 1 0

if ^(f —fo)2+(g—go)2 < 6, otherwise

is the circle function. For effective sources of uniform intensity, each transmission cross-coefficient is proportional to the overlapping area of the effective source and the two displaced pupils. The integral of Eq. (D. 1) can be computed analytically [132,133]. With circular sources, for example, we can determine the overlapping area of three circles by geometry or by using Stoke's theorem [ 134] (see Exercise 8.4). In the following subsections we present aerial images of representative objects produced by various optical systems.

D.2.1 Chromium-on-glass mask under on-axis illumination Object and imaging configuration: g if

Óx{z) = tf

tbg

Imaging parameters 500 nm 0.5 J(f,g) circ(0,0,0.5) ko NA

(z—npx I < d/2 n E Z, otherwise. Object parameters px 1.5 d 0.5 tfg 1 tb g 0

The image and select intensity values are shown in Fig. D.1.

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(5.5)

Simulator Accuracy Tests

208

ó I

position (ko /NA)

L (

0

ó

-0.50

0.00

-0.25

0.25

PL

0.50

position (A/NA)

intensity 0.702211 0.684056 0.631798 0.551715 0.453194 0.347301 0.245127 0.156188 0.087137 0.040992 0.017003

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

Figure D.1: Coupling intensity distribution of a line -space on a chromium-on-glass mask under partially coherent illumination.

D.2.2 Dipole illumination of attenuated phase-shifting mask Object and imaging configuration:

X0

Object [Eq. (5.5)]

Imaging parameters 500 nm

NA

0.5

J(f,g)

circ(+1/2p ,,0,0.2)

Ax d tfg

-

0.8 0.4 1

tb g -(TC

2 )/( it i 2)

The image and select intensity values are shown in Fig. D.2. position (ao /NA) intensity

I

a) ó

-0.4

-0.2

0.0

0.2

0.4

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

0.605236 0.582200 0.516601 0.418425 0.302618 0.186811 0.088635 0.023035 0.000000

position (A/NA)

Figure D.2: Coupling intensity distribution of a line-space on an attenuated phaseshifting mask under dipole illumination.

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209

Images of M = 1 systems

D.2.3 Equal line-space on alternating phase-shifting mask Object and imaging configuration: 1

if

-2np,1 < (p,- 2),

Ô(î) = -1 if (j3 +Z) < 1- 2nß I < (2px

0



(5.22)

otherwise.

Imaging parameters 500 nm Xo NA 0.5

J(f g)

-i),

Object parameters 0.5 A, 0.25 d

(f,8); circ (0, 0, 0.3 )

z

Onm

This is the scenario investigated in Exercise 5.8. The images and select intensity values under the two illumination configurations are shown in Fig. D.3.

r

T

52 5

-0.25

-0.50

0.25

0.00

0.50

position (Xo/NA)

a=0

a = 0.3

0.00

0.810569

0.189713

0.05 0.10 0.15 0.20

0.733167 0.530525 0.280045 0.077403

0.189713 0.189713 0.189713 0.189713

0.25

0.000000

0.189713

position (2./NA)

Figure D.3: Coupling intensity distributions of an equal line-space on an alternating phase-shifting mask under coherent and partially coherent illumination.

D.2.4 Periodic contacts on chromium-on-glass mask Object and imaging configuration: tfg if

1z-mpx ^
tbg otherwise.

Imaging parameters 500 nm NA 0.5 circ(0,0,0.5) f(f,g)

z

Onm

Object parameters (j3,PY) (1.5,2.0) d

0.5 tgg I tb g 0

The image and select intensity v alues are shown in Fig. D.4. Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms

Simulator Accuracy Tests

210

Figure D.4: Coupling intensity distribution of a periodic contact on a chromium-onglass mask with partially coherent illumination.

D.3 Aberrations Proper implementation of the first four Zernike polynomials Zl, Z2, Z3, and Z4 can be ascertained using the displacement theorem (see §7.2.1 and §7.4). Accuracy of the other terms can be determined when the effective source comprises a finite number of source points, as demonstrated in the following section. i

D.4 Finite number of source points

(pe;

When the functions H(f , g), Mik z), and Mk (pë; z) are not constant, the integral of Eq. (8.4) cannot be calculated exactly in general. However, when the illumination consists of a finite number of source points s

f(ƒ,) = Y asb(.f —.fs,g —gs), s=1

the approximate relation in Eq. (8.8) becomes exact, and we can compute intensity distributions analytically by summing over the S source points.

D.4.1 Resist image with aerial coupling medium Object and imaging configuration:

Xo NA J(f,g) M Zo

Imaging parameters 500 nm 0.5 8(f,g), unpolarized 0.25 O nm

Object [Eq. (5.5)] px 1.5 cl 0.4 tfg 1 tbg 0

We can also perform a crude test using the approximate relation of Eq. (7.15).

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Finite number of source points



211

Wafer stack

Zernike coefficients 5 6 7 8 9 10

0.025 -0.015 -0.010 0.005 0.010 -0.020

11

0.005

layer

index

incident medium photoresist substrate

(1,0) (1.8, 0.02) (4.36, 2.11)

thickness

31

The resist image is shown in Fig. D.5; select intensity values are tabulated in Table D.2. Table D.2: Resist image intensity values of a line-space on a chromium-on-glass mask under on-axis coherent illumination.

z = 0 nm z = 250 nm z = 500 nm z = 750 nm z = 1000 nm

1=-0.50

1=-0.25

i =0.00

1=0.25

1=0.50

0.018448 0.006974 0.010222 0.027010 0.037562

0.439115 0.221670 0.119128 0.247531 0.359286

0.787546 0.404300 0.195747 0.419222 0.631156

0.262546 0.129789 0.078412 0.159002 0.222263

0.030041 0.015850 0.004851 0.026332 0.049751

Figure D.5: Resist image of a line-space on a chromium-on-glass mask under onaxis coherent illumination.

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212



Simulator Accuracy Tests

D.4.2 Immersion imaging Object and imaging configuration: Imaging parameters Xo 193 nm sin 9o bj 0.95 NA 1.3775 S(j+ 1/2ß,g) J(ƒ,g) unpolarized M 0.25 O nm zo

Object [Eq. (5.5)]

px 80 nm d

Px/2

tfg 1 —(7r-

tbg

2)ß(n+2)

Wafer stacka layer

index

incident medium top anti-reflection photoresist bottom anti-reflection substrate

thickness

(1.45, 0) (1.55, 0.0) (1.8, 0.02) (1.72, 0.33) (0.883, 2.778)

35 nm 100 nm 87 nm

'Based on data provided by AZ Electronic Materials.

The resist image is plotted in Fig. D.6; select intensity values are tabulated in Table D.3.

Table D.3: Resist image intensity values of an equal line-space on an attenuated phase-shifting mask under dipole illumination.

z = 0 nm z = 25 nm z = 50 nm z= 75 nm

z = l00 nm

x=Onm

x=20nm

x=40nm

0.700581 0.646065 0.406041 0.658162 0.540058

0.439367 0.396177 0.294911 0.404406 0.334714

0.178153 0.146289 0.183781 0.150649 0.129371

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Finite number of source points



213

Figure D.6: Resist image of an equal line-space on an attenuated phase-shifting mask under dipole illumination.

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Appendix E

Select Refractive Indexes The following table lists refractive indexes of some commonly used materials in integrated circuit fabrication [135]: Material

365nm

248nm

193nm

157nm

13nm

Silicon (Si) Germanium (Ge) Gallium Arsenide (GaAs) Indium Phosphide (InP) Chromium(Cr) Glass (SiO2) Amorphous Silicon (a-Si) Silicon Nitride (Si3N4) Tungsten (W) Platinum (Pt) Gold(Au) Silver (Ag) Copper (Cu) Aluminium (Al) Photoresist

6.522,12.705 4.070,12.579 3.596,12.076 3.299,i2.060 1.40,13.26 1.47453,10 3.90,12.66 2.093, i0 3.39,12.66 1.62,12.62 1.716,11.862 0.186,11.61 1.27,11.95 0.407,14,43 - (1.58,i0.02)

1.68,13.58 1.394,i3.197 2.273,i4.084 2.131,i3.495 0.85,12.01 1.50841,íO 1.69,i2.76 2.278,10.005 3.40,12.85 1.36,11.76 1.484,11.636 1.298,11.35 1.47,11.78 0.19,12.94 " (1.65,i0.02) 1í

0.883,12.778 1.13,12.09 1.358,i2.013 1.485,i2.008 0.84,ií.65 1.563,iO 0.974,i2.10 2.66,iO.243 1.30,i3.02 1.316,11.28 1.425,11.156 1.028,11.18 0.97,il.40 0.113,i2.21 . (1.8,i0.02)"í

0.494,12.04 0.928,11.44 0.916,11.476 0.973,il.594 0.676,11.11 1.689,íO 0.675,ií.63 2.675,10.923 0.932,12.00 1.46,11.18 1.463,11.107 0.951,10.738 1.026,ií.04 0.0736,11.69 ,- (1.6,10.02)

1.03,1113 0.96,iO.027 1.0,iO.0307 —

0.938,íO.035 0.9789,17.6 — — — —

0.889,10.05 0.862,10.055 0.965,iO.069 —

-' (0.98,i0.005) 1°

Kodak KTFR photoresist [136]. Kodak KTFR photoresist [136]. Data provided by AZ Electronic Materials. ' v From http://www.cxro.lbl.gov/optical_constants.

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Appendix F

Assorted Theorems and Identities Divergence theorem (Gauss's law):

f

J a•ndS= v

S

V •adV.

Stoke's theorem:

ic a - idl

=I

Vxa•ndS.

S

Green's theorem:

fff

(UVG— GV Z U) dV

v

= ff

án—

G-

1

dS.

S

Select identities: V • V x a - 0.

(Exercise 1.1)

V x V f = 0.

(Exercise 1.2)

—V.(axb)-a•Vxb—b•Vxa.

(Exercise 1.3)

A x (B x C) - (A • C)B — (A B)C.

V • (UVG) = UV 2 G+ VU. VG.

(Exercise 3.1)

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Bibliography [1] J. A. Stratton, Electromagnetic Theory, McGraw-Hill (1941). [2] J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, 2nd ed. (1975). [3] L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, Pergamon Press, 2nd ed. (1984). [4] J. A. Stratton, Electromagnetic Theory, McGraw-Hill, pp. 16-23 (1941). [5] R. F. Harrington, Time-harmonic Electromagnetic Fields, McGraw-Hill, Inc. (1961). [6] M. Born and E. Wolf, Principles of Optics, Pergamon Press, sixth ed. (1980). [7] W. Smith, Modern optical engineering, McGraw-Hill, 2nd ed., pp. 301-346 (1990). [8] W. J. Smith, Modern optical engineering, McGraw-Hill, pp. 135-139 (1966). [9] A. Sommerfeld, Partial differential equations in physics, Academic Press, pp. 188-200 (1949). [10] G. C. Sherman, "Application of the convolution theorem to Rayleigh's integral formulas," J. Opt. Soc. Am., vol. 57, pp. 546-547 (1967). [11] J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill, pp. 42-46 (1968). [12] G. B. Airy, "On the diffraction of an object-glass with circular aperture," Transactions of the Cambridge Philosophical Society, vol. 5, no. 3, pp. 283291 (1835). [13] J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill (1968).

219

Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms

220



Bibliography

[14] D. G. Flagello and A. E. Rosenbluth, "Lithographic tolerances based on vector diffraction theory," J. Vac. Sci. Technol. B, vol. 10, no. 6, pp. 2997-3003 (1992). [15] G. M. Gallatin, "High numerical aperture scalar imaging," Applied Optics, vol. 40, no. 28, pp. 4958-4964 (2001). [16] D. C. O'Shea, Elements of modern optical design, John Wiley & Sons, pp. 111-114 (1985). [17] M. Born and E. Wolf, Principles of Optics, Pergamon Press, sixth ed., pp. 522-524 (1980). [18] H. Hopkins, "Canonical coordinates in geometrical and diffraction image theory," Jpn. J. Appl. Phys., vol. 4, no. 1, pp. 31-35 (1965). [19] M. Born and E. Wolf, Principles of Optics, Pergamon Press, sixth ed., pp. 528-532 (1980). [20] Rayleigh, "Investigations in optics, with special reference to the spectroscope," The Philosophical Magazine, vol. 8, no. 49, pp. 261-274 (1879). [21] M. Born and E. Wolf, Principles of Optics, Pergamon Press, sixth ed., pp. 414-418 (1980). [22] D. Halliday and R. Resnick, Physics, John Wiley & Sons, extended ed., pp. 1034-1037 (1986). [23] A. K. Wong, Resolution Enhancement Techniques in Optical Lithography, SPIE Press (2001). [24] G. O. Reynolds, "A concept for a high resolution optical lithographic system for producing one-half micron linewidths," in H. L. Stover, ed., Proc. SPIE, vol. 633, pp. 228-238 (1986). [25] K. Kamon, T. Miyamoto, Y. Myoi, H. Nagata, M. Tanaka, and K. Horie, "Photolithography system using annular illumination," Jpn. J. Appl. Phys., vol. 30, no. 11B, pp. 3021-3029 (1991). [26] M. Noguchi, M. Muraki, Y. Iwasaki, and A. Suzuki, "Subhalf micron lithography system with phase-shifting effect," in J. D. Cuthbert, ed., Proc. SPIE, vol. 1674, pp. 92-104 (1992). [27] H.-Y. Kang, C.-H. Kim, J.-H. Lee, W.-S. Han, and Y.-B. Koh, "High performance lithography with advanced modified illumination," IEEE Transactions on Electronics, vol. 77-C, no. 3, pp. 432-437 (1994).

Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms

Bibliography



221

[28] T. Ogawa, M. Uematsu, T. Ishimaru, and M. Kimura, "The effective light source optimization with the modified beam for the depth-of-focus enhancements," in T. Brunner, ed., Proc. SPIE, vol. 2197, pp. 19-30 (1994). [29] B. J. Lin, "The attenuated phase-shifting mask," Solid State Technology, vol. 35, no. 1, pp. 43-47 (1992). [30] T. Terasawa, N. Hasegawa, and H. Fukuda, "Imaging characteristics of multi-phase-shifting and halftone phase-shifting masks," Jpn. J. Appl. Phys., vol. 30, no. I 1B, pp. 2991-2997 (1991). [31] M. Born and E. Wolf, Principles of Optics, Pergamon Press, sixth ed., pp. 265-268 (1980). [32] M. Levenson, N. Viswanathan, and R. Simpson, "Improving resolution in photolithography with a phase-shifting mask," IEEE Transactions on Electron Devices, vol. 29, no. 12, pp. 1812-1846 (1982). [33] L. Liebmann, I. Graur, W. Leipold, J. Oberschmidt, D. O'Grady, and D. Regaill, "Alternating phase shifted mask for logic gate levels, design and mask manufacturing," in L. van den Hove, ed., Proc. SPIE, vol. 3679, pp. 27-37 (1999). [34] H.-Y. Liu, L. Karklin, Y.-T. Wang, and Y. C. Pati, "The application of alternating phase-shifting masks to 140 nm gate patterning: Line width control improvements and design optimization," in Proc. SPIE, vol. 3236, pp. 328337 (1998). [35] S. Nakao, A. Nakae, K. Tsujita, and W. Wakamiya, "Impact of spherical aberrations on printing characteristics of irregularly aligned patterns of alternating phase shift mask," Jpn. J. Appl. Phys., vol. 38, no. 4A, pp. 1919-1926 (1999). [36] R. Schenker, "Effects of phase shift masks on across field linewidth control," in L. van den Hove, ed., Proc. SPIE, vol. 3679, pp. 18-26 (1999). [37] M. D. Levenson, "Extending the lifetime of optical lithography technologies with wavefront engineering," Jpn. J. Appl. Phys., vol. 33, no. 12B, pp. 67656773 (1994). [38] B. J. Lin, "Partially coherent imaging in two-dimensions and theoretical limits of projection printing in microfabrication," IEEE Transactions on Electron Devices, vol. 27, no. 5, pp. 931-938 (1980). [39] F. M. Schellenberg, Selected Papers on Resolution Enhancement Techniques in Optical Lithography, SPIE Press (2004).

Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms

222



Bibliography

[40] T. Toyoshima, T. Ishibashi, A. Minanide, K. Sugino, K. Katayama, T. Shoya, I. Arimoto, N. Yasuda, H. Adachi, and Y. Matsui, "0.1µm level contact hole pattern formation with KrF lithography by resolution enhancement lithography assisted by chemical shrink (RELACS)," in Technical Digest— International Electron Devices Meeting, pp. 333-336 (1998). [41] A. Yariv, Quantum electronics, John Wiley & Sons, 3rd ed., pp. 663-665 (1989). [42] M. S. Yeung, D. Lee, R. Lee, and A. R. Neureuther, "Extension of the Hopkins theory of partially coherent imaging to include thin-film interference effects," in J. Cuthbert, ed., Proc. SPIE, vol. 1927, pp. 452-463 (1993). [43] D. G. Flagello, High Numerical Aperture Imaging in Homogeneous Thin Films, Ph.D. thesis, University of Arizona (1993). [44] D. G. Flagello, T. Milster, and A. E. Rosenbluth, "Theory of high-NA imaging in homogeneous thin films," J. Opt. Soc. Am. A, vol. 13, no. 1, pp. 53-64 (1996). [45] P. H. Berning, "Theory and calculations of optical thin films," in G. Hass, ed., Physics of Thin Films, vol. 1, Academic Press, pp. 69-121 (1963). [46] T. A. Brunner, "Optimization of optical properties of resist processes," in H. Ito, ed., Proc. SPIE, vol. 1466, pp. 297-308 (1991). [47] B. J. Lin, ed., Journal of Microlithography, Microfabrication, and Microsystems, vol. 3(1), SPIE Press, pp. 8-114 (2004). [48] V. N. Mahajan, Aberration Theory Made Simple, SPIE Press (1991). [49] F. Zernike, "Beugungstheorie des schneidenverfahrens und seiner verbesserten form, der phasenkontrastmethode," Physica, vol. 1, pp. 689704 (1934). [50] F. Zernike, "Diffraction theory of the knife-edge test and its improved form, the phase-contrast method," JM3 , vol. 1, no. 2, pp. 87-94 (2002). [51] M. Born and E. Wolf, Principles of Optics, Pergamon Press, sixth ed., pp. 203-207 (1980). [52] D. Malacara, Optical shop testing, John Wiley & Sons (1978). [53] D. G. Flagello, R. J. Socha, X. Shi, J. B. van Schoot, J. Baselmans, M. A. van de Kerkhof, W. de Boeij, A. Engelen, R. Carpaij, O. Noordman, M. H. Moers, M. Mulder, J. Finders, H. van Greevenbroek, M. Schriever, M. Maul, H. Haidner, M. Goeppert, U. Wegmann, and P. Graeupner, "Optimizing and

Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms

Bibliography



223

enhancing optical systems to meet the low kl challenge," in A. Yen, ed., Proc. SPIE, vol. 5040, pp. 139-150 (2003). [54] M. A. van de Kerkhof, W. de Boeij, H. Kok, M. Silova, J. Baselmans, and M. Hemerik, "Full optical column characterization of DUV lithographic projection tools," in B. Smith, ed., Proc. SPIE, vol. 5377, pp. 1960-1970 (2004). [55] P. Dirksen, J. Braat, A. J. E. M. Janssen, and C. Juffermans, "Aberration retrieval using the extended Nijboer-Zernike approach," JM3 , vol. 2, no. 1, pp. 61-68 (2003). [56] D. Malacara, Optical shop testing, John Wiley & Sons, pp. 323-349 (1978). [57] N. R. Farrar, A. H. Smith, D. R. Busath, and D. Taitano, "In-situ measurement of lens aberrations," in C. Progler, ed., Proc. SPIE, vol. 4000, pp. 18-29 (2000). [58] G. Robins, K. Adam, and A. R. Neureuther, "Measuring optical image aberrations with pattern and probe based targets," J. Vac. Sci. Technol. B, vol. 20, no. 1, pp. 338-343 (2002). [59] G. Robins and A. R. Neureuther, "Operational model for pattern and probe based aberration monitors," J. Vac. Sci. Technol. B, vol. 20, no. 6, pp. 25102616 (2002). [60] M. Born and E. Wolf, Principles of Optics, Pergamon Press, sixth ed., pp. 424-428 (1980). [61] K. Toh, Two-dimensional Images with Effects of Lens Aberrations in Optical Lithography, Master's thesis, University of California, Berkeley, memorandum No. UCB/ERL M88/30 (1988). [62] H. Gamo, "Matrix treatment of partial coherence," in E. Wolf, ed., Progress in Optics, vol. 3, North-Holland, pp. 187-332 (1964). [63] Y. C. Pati, A. A. Ghazanfarian, and R. F. Pease, "Exploiting structure in fast aerial image computation for integrated circuit patterns," IEEE Transactions on Semiconductor Manufacturing, vol. 10, no. 1, pp. 62-74 (1997). [64] A. E. Rosenbluth, G. Gallatin, R. Gordon, W. Hinsberg, J. Hoffnagle, F. Houle, K. Lai, A. Lvov, M. Sanchez, and N. Seong, "Fast calculation of images for high numerical aperture lithography," in B. Smith, ed., Proc. SPIE, vol. 5377, pp. 615-628 (2004). [65] K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Transactions on Antennas Propagation, vol. 14, pp. 302-307 (1966). Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms

224

Bibliography

[66] R. Guerrieri, K. H. Tadros, J. Gamelin, and A. Neureuther, "Massively parallel algorithms for scattering in optical lithography," IEEE Transactions on Computer-aided Design of Integrated Circuits and Systems, vol. 10, no. 9, pp. 1091-1100 (1991). [67] A. Wong and A. Neureuther, "Rigorous three-dimensional time-domain finite-difference electromagnetic simulation for photolithographic applications," IEEE Transactions on Semiconductor Manufacturing, vol. 8, no. 4, pp. 419-431 (1995). [68] M. S. Yeung, "Fast and rigorous three-dimensional mask diffraction simulation using Battle-Lemarie wavelet-based multiresolution time-domain method," in A. Yen, ed., Proc. SPIE, vol. 5040, pp. 69-77 (2003). [69] G. L. Wojcik, J. Mould, R. Ferguson, R. Martino, and K. K. Low, "Some image modeling issues for i -line 5x phase shifting masks," in T. Brunner, ed., Proc. SPIE, vol. 2197, pp. 455-465 (1994). [70] D. Nyyssonen and C. P. Kirk, "Optical microscope imaging of lines patterned in thick layers with variable edge geometry: theory," J. Opt. Soc. Am. A, vol. 5, no. 8, pp. 1270-1280 (1988). [71] K. Lucas, C.-M. Yuan, and A. Strojwas, "A rigorous and practical vector model for phase shifting masks in optical lithography," in J. D. Cuthbert, ed., Proc. SPIE, vol. 1674, pp. 252-263 (1992). [72] T. Matsuzawa, A. Moniwa, N. Hasegawa, and H. Sunami, "Two-dimensional simulation of photolithography on relective stepped substrate," IEEE Transactions on Computer-aided Design of Integrated Circuits and Systems, vol. 6, no. 3, pp. 446-451 (1987).

[73] E. Barouch, B. Bradie, G. Karniadakis, and S. Orszag, "Comprehensive 3D simulator with non-planar substrates," in V. Pol, ed., Proc. SPIE, vol. 1264, pp. 334-342 (1990). [74] J. Bischoff, U. Glaubitz, and N. Haase, "New method of topography simulation in photolithography," in J. D. Cuthbert, ed., Proc. SPIE, vol. 1674, pp. 423-434 (1992). [75] M. Yeung and E. Barouch, "Three-dimensional nonplanar lithography simulation using a periodic fast multipole method," in G. Fuller, ed., Proc. SPIE, vol. 3051, pp. 509-521 (1997). [76] J.-P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," Journal of Computational Physics, vol. 114, pp. 185-200 (1994). Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms

Bibliography



225

[77] J.-P. Berenger, "Three-dimensional perfectly matched layer for the absorption of electromagnetic waves," Journal of Computational Physics, vol, 127, pp. 363-379 (1996). [78] H. Kirchauer and S. Selberherr, "Rigorous three-dimensional photoresist exposure and development simulation over nonplanar topography," IEEE Transactions on Computer-aided Design of Integrated Circuits and Systems, vol. 16, no. 12, pp. 1431-1438 (1997).

[79] A. Erdmann, C. Kalus, T. Schmoeller, Y. Klyonova, T. Sato, A. Endo, T. Shibata, and Y. Kobayashi, "Rigorous simulation of exposure over nonplanar wafers," in A. Yen, ed., Proc. SPIE, vol. 5040, pp. 101-111 (2003). [80] M. Zuniga, N. Rau, and A. Neureuther, "Application of a general reaction/diffusion resist model to emerging materials with extension to non actinic exposure," in Proc. SPIE, vol. 3049, pp. 256-268 (1997). [81] A. Erdmann, W. Henke, S. Robertson, E. Richter, B. Tollkühn, and W. Hoppe, "Comparison of simulation approaches for chemically amplified resists," in C. A. Mack and T. Stevenson, eds., Proc. SPIE, vol. 4404, pp. 99110 (2001). [82] H. Fukuda, K. Hattori, and H. Takuya, "Impact of acid/quencher behavior on lithography performance," in C. Progler, ed., Proc. SPIE, vol. 4346, pp. 319330 (2001). [83] W. Hinsberg, F. A. Houle, and H. Ito, "Reactive dissolution kinetics of lithographic copolymers," in J. L. Sturtevant, ed., Proc. SPIE, vol. 5376, pp. 352359 (2004). [84] J. A. Sethian, "Fast marching level set methods for three-dimensional photolithography development," in G. Fuller, ed., Proc. SPIE, vol. 2726, pp. 262-272 (1996). [85] T. Brunner and R. Ferguson, "Simple models for resist processing effects," Solid State Technology, pp. 95-96,98,100,103 (1996). [86] C. A. Mack, "Enhanced lumped parameter model for photolithography," in Proc. SPIE, vol. 2197, pp. 501-510 (1994). [87] N. Cobb and A. Zakhor, "Fast sparse aerial-image calculation for OPC," in G. V. Shelden and J. N. Wiley, eds., Proc. SPIE, voL 2621, pp. 534-545 (1995). [88] B. Tollkühn, A. Erdmann, J. Lammers, C. Nolscher, and A. Semmler, "Do we need complex resist models for predictive simulation of lithographic process performance?" in J. L. Sturtevant, ed., Proc. SPIE, vol. 5376, pp. 983994 (2004). Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms

226

Bibliography

[89] M. D. Levenson, "The phase-shifting mask II: Imaging simulations and submicrometer resist exposures," IEEE Transactions on Electron Devices, vol. 31, no. 6, pp. 753-763 (1984). [90] A. Wong, R. Ferguson, S. Mansfield, A. Molless, D. Samuels, R. Schuster, and A. Thomas, "Level-specific lithography optimization for 1 Gb DRAM," IEEE Transactions on Semiconductor Manufacturing, vol. 13, no. 1, pp. 7687 (2000). [91] A. Wong, R. Ferguson, and S. Mansfield, "The mask error factor in optical lithography," IEEE Transactions on Semiconductor Manufacturing, vol. 13, no. 2, pp. 235-242 (2000). [92] Y. Granik and N. B. Cobb, "MEEF as a matrix," in G. T. Dao and B. J. Grenon, eds., Proc. SPIE, vol. 4562, pp. 980-991 (2001). [93] A. Wong, R. Ferguson, L. Liebmann, S. Mansfield, A. Molless, and M. Neisser, "Lithographic effects of mask critical dimension error," in L. van den Hove, ed., Proc. SPIE, vol. 3334, pp. 106-116 (1998). [94] R. Kostelak, C. Pierrat, J. Garofalo, and S. Vaidya, "Exposure characteristics of alternate aperture phase-shifting masks fabricated using a subtractive process," J. Vac. Sci. Technol. B, vol. 10, no. 6, pp. 3055-3061 (1992). [95] C. Pierrat, A. Wong, and S. Vaidya, "Phase-shifting mask topography effects on lithographic image quality," in Technical Digest—International Electron Devices Meeting, pp. 53-56 (1992). [96] A. Wong and A. Neureuther, "Mask topography effects in projection printing of phase-shifting masks," IEEE Transactions on Electron Devices, vol. 41, no. 6, pp. 895-902 (1994). [97] A. K. Wong and A. R. Neureuther, "Polarization effects in mask transmission," in J. D. Cuthbert, ed., Proc. SPIE, vol. 1674, pp. 193-200 (1992). [98] A. K. Wong and C. Pierrat, "Wave-optical considerations in optical lithography," in F. Wyrowski, ed., Proc. SPIE, vol. 5182, pp. 63-72 (2003). [99] J. Kirk and C. Progler, "Pupil illumination: In-situ measurement of partial coherence," in L. van den Hove, ed., Proc. SPIE, vol. 3334, pp. 281-288 (1998). [100] T. Brunner, "Impact of lens aberrations on optical lithography," IBM J. Res. Develop., vol. 41, no. 1/2, pp. 57-67 (1997). [101] C. Progler and D. Wheeler, "Optical lens specifications from the users' perspective," in L. van den Hove, ed., Proc. SPIE, vol. 3334, pp. 256-268 (1998). Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms

Bibliography

227

[102] D. G. Flagello, J. de Klerk, G. Davies, and R. Rogoff, "Towards a comprehensive control of full-field image quality in optical lithography," in G. Fuller, ed., Proc. SPIE, vol. 3051, pp. 672-685 (1997). [103] M. Born and E. Wolf, Principles of Optics, Pergamon Press, sixth ed., pp. 468-473 (1980). [104] B. J. Lin, "Where is the lost resolution?" in H. L. Stover, ed., Proc. SPIE, vol. 633, pp. 44-50 (1986).

[ 105] J. Kirk, "Astigmatism and field curvature from pin-bars," in V. Pol, ed., Proc. SPIE, vol. 1463, pp. 282-291 (1991). [ 106] T. A. Brunner, A. L. Martin, R. M. Martion, C. P. Ausschnitt, and T. H. New-

man, "Quantitative stepper metrology using the focus monitor test mask," in T. Brunner, ed., Proc. SPIE, vol. 2197, pp. 541-549 (1994). [107] A. Starikov, "Exposure monitor structure," in Proc. SPIE, vol. 1261, pp. 315-324 (1990). [ 108] C. P. Ausschnitt and M. E. Lagus, "Seeing the forest for the trees: a new approach to CD control," in Proc. SPIE, vol. 3332, pp. 212-220 (1998).

[109] C. P. Ausschnitt, "Distinguishing dose from defocus for in-line lithography control," in B. Singh, ed., Proc. SPIE, vol. 3677, pp. 140-147 (1999). [110] J. C. Stover and M. L. Bernt, "Measurement and analysis of scatter from rough surfaces," in R. P. Breault, ed., Proc. SPIE, vol. 2260, pp. 40-49 (1994).

[111] J. Kirk, "Scattered light in photolithographic lenses," in T. Brunner, ed., Proc. SPIE, vol. 2197, pp. 566-572 (1994). [112] C. Progler and A. Wong, "Zernike coefficients: are they really enough?" in C. Progler, ed., Proc. SPIE, vol. 4000, pp. 40-52 (2000). [113] A. K. Wong, A. F. Molless, T. A. Brunner, E. Coker, R. H. Fair, G. L.

Mack, and S. M. Mansfield, "Linewidth variation characterization by spatial decomposition," Journal of Microlithography, Microfabrication, and Microsystems, vol. 1, no. 2, pp. 106-116 (2002). [ 114] A. K. Wong, A. F. Molless, T. A. Brunner, E. Coker, R. H. Fair, G. L.

Mack, and S. M. Mansfield, "Characterization of line width variation," in C. Progler, ed., Proc. SPIE, vol. 4000, pp. 184-191 (2000).

Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms

228

Bibliography

[115] P. Vanoppen, O. Noordman, J. Baselmans, and J. van Schoot, "Analysis of full wafer/full batch CD uniformity using electrical line width measurements," in C. Mack and T. Stevenson, eds., Proc. SPIE, vol. 4404, pp. 33-44 (2001).

[116] "Techniques on microphotography," Kodak sales service publication P -52 (1963). [117] M. D. Prouty and A. R. Neureuther, "Optical imaging with phase shift masks," in Proc. SPIE, vol. 470, pp. 228-232 (1984). [118] J. Garofalo, C. Biddick, R. Kostelak, and S. Vaidya, "Mask assisted off axis illumination technique for random logic," J. Vac. Sci. Technol. B, vol. 11, no. 6, pp. 2651-2658 (1993). [ 119] J. Garofalo, 0. Otto, R. Cirelli, R. Kostelak, and S. Vaidya, "Automated layout of mask assist-features for realizing 0.5k1 ASIC lithography," in T. Brunner, ed., Proc. SPIE, vol. 2440, pp. 302-312 (1995).

[120] J. F. Chen, T. Laidig, K. Wampler, and R. Caldwell, "Optical proximity correction for intermediate-pitch features using sub-resolution scattering bars," J. Vac. Sci. Technol. B, vol. 15, no. 6, pp. 2426-2433 (1997). [121] O. Otto and R. Henderson, "Advances in process matching for rules-based optical proximity correction," in Proc. SPIE, vol. 2884, pp. 425-434 (1996). [ 122] N. Cobb and A. Zakhor, "Experimental results on optical proximity correction with variable threshold resist model," in G. Fuller, ed., Proc. SPIE, vol. 3051, pp. 458-468 (1997). [123] M. Rieger and J. Stirniman, "Mask fabrication rules for proximity corrected patterns," in Proc. SPIE, vol. 2884, pp. 323-332 (1996). [ 124] T. Waas, H. Eisenmann, and H. Hartmann, "Proximity correction for high CD-accuracy and process tolerance," in Proc. Symposium on Nanocircuit Engineering (1994).

[125] H.-Y. Liu, L. Karklin, Y.-T. Wang, and Y. C. Pati, "Application of alternating phase-shifting masks to 140 nm gate patterning II: Mask design and manufacturing tolerances," in Proc. SPIE, vol. 3334, pp. 2-14 (1998). [126] A. N. Neureuther and C. G. Willson, "Reduction in x-ray lithography shot noise exposure limit by dissolution phenomena," J. Vac. Sci. Technol. B, vol. 6, no. 1, pp. 167-173 (1988). [127] G. M. Gallatin, "Continuum model of shot noise and line edge roughness," in C. A. Mack, ed., Proc. SPIE, vol. 4404, pp. 123-132 (2001).

Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms

Bibliography

229

[ 128] R. L. Brainard, P. Trefonas, J. H. Lammers, C. A. Cutler, J. F. Mackevich, A. Trefonas, and S. A. Robertson, "Shot noise, LER, and quantum efficiency of EUV photoresists," in R. S. Mackay, ed., Proc. SPIE, vol. 5374, pp. 74-85 (2004). [129] M. Born and E. Wolf, Principles of Optics, Pergamon Press, sixth ed., pp. 665-718 (1980). [ 130] A. E. Rosenbluth, G. Gallatin, N. Seong, O. Dittmann, and M. Totzeck, "Image formation in a lens with spatial dispersion," in Proc. 4 th International Symposium on 157 -nm Lithography (2003).

[131] N. Seong, K. Lai, A. E. Rosenbluth, and G. Gallatin, "Assessing the impact of intrinsic birefringence on 157 -nm lithography," in B. Smith, ed., Proc. SPIE, vol. 5377, pp. 99-103 (2004). [132] E. C. Kintner, "Method for the calculation of partially coherent imagery," Applied Optics, vol. 17, no. 17, pp. 2747-2753 (1978). [ 133] S. Subramanian, "Rapid calculation of defocused partially coherent images," Applied Optics, vol. 20, no. 10, pp. 1854-1857 (1981).

[ 134] R. L. Gordon and A. E. Rosenbluth, "Lithographic image simulation for the 21 th century with 19 t -century tools," in F. Wyrowski, ed., Proc. SPIE, vol. 5182, pp. 73-87 (2003). [135] E. D. Palik, ed., Handbook of optical constants of solids, Academic Press (1985). [ 136] A. J. Warnecke and P. J. Lopresti, "Refractive index dispersion in semiconductor-related thin films," IBM J. Res. Develop., pp. 256-262 (1973).

Downloaded from SPIE Digital Library on 25 Feb 2010 to 130.60.68.45. Terms of Use: http://spiedl.org/terms

1 ' /

balancing, see aberration, balancing Bessel function, 47 bias mask order equalization, 95 process, 94, 170 binary intensity object, 78 birefringence, see refraction, double form, 195 stress, 195 Brewster angle, 115

Abbe's method, see source points integration aberration astigmatism, 141 balancing, 139 coma, 142 defocus, see defocus displacement theorem, 135-136 piston, 141 Seidel theory, 133 shamrock, 142 spherical, 143 tilt, 141 absorption coefficient, 14 aerial image, 107 Airy pattern, see diffraction, Fraunhofer, circular aperture alternating phase-shifting mask, see resolution enhancement, alternating phase-shifting mask angular frequency, see frequency, angular aquatic image, 107 astigmatism, see aberration, astigmatism attenuated phase-shifting mask, see resolution enhancement, attenuated phase-shifting mask attenuation index, see extinction coefficient auto-coherence function, see coherence, spatial, auto-coherence function auto-correlation function, 73

canonical coordinates, 67 characteristic impedance, 8 characteristic matrix, 120 transverse electric, 120 transverse magnetic, 121 circle function, 207 circular polarization, see polarization, circular coherence spatial, 58-64 auto-coherence function, 59 coherent, 60 complex degree of coherence, see complex degree of coherence incoherent, 60 mutual intensity, see mutual intensity partially coherent, 60 temporal, 58 coherence length, 17 coherence time, 17, 197

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250



Index

quasi-monochromatic light, 1418 coherence length, see coherence, tem poral, coherence length coherence time, see coherence, temporal, coherence time coherency matrix, see polarization, coherency matrix coherent decomposition, 157-159 coma, see aberration, coma complex correlation factor, see polarization, complex correlation factor complex degree of coherence, 60, 61 Köhler illumination, 66 quasi-monochromatic light, 60 condenser lens, 65 conductivity, 3 constitutive relations, 3 continuity equation, 2 contrast, 87 coupling image, 107 defocus, 141, 179-181 Rayleigh's unit, 180 degree of first-order modulation, 85 alternating phase-shifting mask, 88 attenuated phase-shifting mask, 86 off-axis illumination, 85 degree of polarization, see polarization, degree of dielectric constant, see permittivity dielectric tensor, 191 diffraction Fraunhofer, 41-44 annular aperture, 46-49 circular aperture, 46-49, 93 rectangular aperture, 45-46 relationship with transfer function, 55 scaling, 49 Fresnel, 35-39

pattern, see point spread function Rayleigh- Sommerfeld, 39-41 diffraction focus, 135 direction cosines, 103 dispersion, 17 displacement theorem, see aberration, displacement theorem divergence theorem, 18 dose to clear, 167, 185 dose to harden, 167 dose to print, see exposure dose dose variation, 182-183 double refraction, see refraction, double edge roughness, see mask-edge roughness or line-edge roughness effective source, 68 eikonal, 20 eikonal equation, 20 elliptic polarization, see polarization, elliptic energy density, 4 geometrical optics, 20 ergodicity, 197 evanescent wave, 116 exposure dose, 167, 185 extinction coefficient, 13 feature, 165 flare, 183-186 flare-amplification factor, 186 fluctuation, 165 focal length, 26 focal plane, 26 focus, see defocus; diffraction focus Fraunhofer diffraction, see diffraction, Fraunhofer

frequency, 10 angular, 10 Fresnel diffraction, see diffraction, Fres• nel

Fresnel's formulae, 111-112

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Index



251

fringe visibility, see contrast Fringe Zernike, see Zernike polynomial, Fringe Zernike convention Gauss's law, 18 Green's function, 34 Green's theorem, 49 group velocity, see velocity, group half-pitch minimum, 75, 89-90 alternating phase-shifting mask, 89 coherent illumination, 80 general, 77 off-axis illumination, 84 partially coherent illumination, 83 harmonic wave, see wave, harmonic Hartmann test, 146-147 Hopkins' approach, see transmission cross-coefficient Huygens-Fresnel principle, 37 illumination critical, 65 Köhler, 65-67 immersion lithography, 126-127 incoherent, see coherence, spatial, incoherent inherent variation, 165 intensity definition, 5 normalization, 55-56 internal reflection, see total internal reflection k1, 94

lithography difficulty, 165 mask-error factor, 171 proximity effect, 170 k1 half-pitch' 90

lithography difficulty, 165

mask-error factor, 171 proximity effect, 170 k2, 181 Köhler illumination, see illumination, Köhler

Kirchhoff's approximations, 36 latent image, 126 law of reflection, 109 law of refraction, see Snell's law line spread function, 172 line-edge roughness photoresist, 189 linear polarization, see polarization, linear Lorentz force equation, 3, 97 magnification angular, 28 immersion, 127 lateral, 26 immersion, 127 longitudinal, 28 immersion, 127 mask-edge roughness, 175 mask-error enhancement factor, see mask error factor mask-error factor, 170-173 material equations, see constitutive relations Maxwell's equations, 2 integral form. 2 mean-square wavefront deformation, 137 meridional plane, 103 Michelson interferometer, 144 minimum dimension, 75, 90-94 minimum half-pitch, see half-pitch, minimum minimum pitch, see half-pitch, minimum mutual coherence function, 59 quasi-monochromatic light, 60

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252



Index

mutual intensity, 61 Nijboer-Zernike method, 144-146 numerical aperture, 54 obliquity factor, 56-58 immersion, 127 off-axis illumination, see resolution enhancement, off-axis illumination Ohm's law, 3 optical and process correction, see resolution enhancement, optical and process correction optical path, see eikonal optical proximity correction, see resolution enhancement, optical and process correction p-polarization, see polarization, transverse magnetic paraxial ray, 24 partial coherence factor, 68 partially coherent, see coherence, spatial, partially coherent periodic space, 78 permeability, 3 permittivity, 3 principal, 194 phase error, 174-175 phase velocity, see velocity, phase phase-contrast method, 149 phase-edge mask, see resolution enhancement, phase-edge mask phase-shifting mask, see resolution enhancement phasor, 11, 12 photoresist roughness, see line-edge roughness, photoresist physical optics, see diffraction piston, see aberration, piston pitch, see half-pitch plane of incidence, 108

plane wave, see wave, plane point spread function, 44, see also diffraction polarization, 97-102 circular, 98 coherency matrix, 99 complex correlation factor, 99 degree of, 98, 101, 177 upon reflection and transmission, 114 elliptic, 98 linear, 98 parallel, see polarization, transverse magnetic perpendicular, see polarization, trans verse electric transverse electric, 103, 117 image, 128 transverse magnetic, 103, 117 image, 128 polarization-induced stray light, 107, 128 polarizing angle, see Brewster angle Poynting vector, 4 geometrical optics, 20 harmonic wave, 12 plane wave, 8 Poynting's theorem of energy conservation, 4 principal dielectric constant, see permittivity, principal principal permittivity, see permittivity, principal principal propagation velocity, see velocity, principal propagation process bias, see bias, process projection lens, 26 proximity effect, 167-170 pupil, 44 pupil function, 52, see also transfer function, coherent aberration-free, 53

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Index



253

quadrant function, 158 quarter-wave plate, 243 quasi-monochromatic light, see coherence, temporal, quasi-monochromatic light radiation condition, see Sommerfeld radiation condition ray aberration, 140 Rayleigh's resolution, see resolution, Rayleigh's criterion Rayleigh's unit, see defocus, Rayleigh's unit Rayleigh-Sommerfeld diffraction, see diffraction, Rayleigh-Sommerfeld reciprocity, 33-34 scalar field, 34 vector field, 34 reduction system, 28 reflection, 108-109 reflection coefficient, 111-112, 122123 reflectivity, 113 refraction, 108-109 double, 195 refractive index, 6 relaxation time, 233 resist image, 107 resolution, see also half-pitch Rayleigh's criterion, 75, 93 resolution enhancement alternating phase-shifting mask, 87, 89 attenuated phase-shifting mask, 8587, 94 off-axis illumination, 83-84 optical and process correction, 188 phase-edge mask, 91 s-polarization, see polarization, transverse electric Seidel theory of aberrations, see aberration, Seidel theory

shamrock, see aberration, shamrock shot noise, 188 sinc function, 45 Snell's law, 23, 109 Sommerfeld radiation condition, 36 source points integration, 155-157 speed, see velocity spherical aberration, see aberration, spherical spherical wave, see wave, spherical stationarity, 197 Stoke's theorem, 18 stratified medium, 116 Strehl ratio, 135 telecentricity error, 84, 177 thin lens, 24-26 thin-mask approximation, see thin-object approximation thin-object approximation, 159 threshold intensity, 167, 188 relation to exposure dose, 167 tilt, see aberration, tilt time-domain finite-difference, 160, 174 total internal reflection, 115-116 transfer function coherent, 51, see also pupil function incoherent, 73 relationship with Fraunhofer diffraction pattern, 55 with obliquity factor, 57 transmission coefficient, 111-112,122123 transmission cross-coefficient, 70 computation, 153-155 transmission error, 174-175 transmissivity, 113 transverse electric, see polarization, transverse electric transverse magnetic, see polarization, transverse magnetic

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254



Index

van Cittert-Zernike theorem, 63 Köhler illumination, 67 velocity electromagnetic wave, 6 group, 16, 234 phase, 234 principal propagation, 194 vignetting, 65, 179 wave equation, 5 harmonic, 9-14 plane, 6-8 spherical, 9 wave equation, see wave, equation wave number, 10 wavelength, 10 Zernike phase-contrast method, see phasecontrast method Zernike polynomial, 137-140 Fringe Zernike convention, 140

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