correction masks for thickness uniformity in large-area thin films

Correction masks for thickness uniformity inlarge-area thin films

Francisco Villa, Amalia Martınez, and Luis E. Regalado

Experimental results from the emission of vapor sources are considered in designing correcting dia-phragms to achieve a uniform thickness distribution during evaporation of thin films mounted onlarge-area substrate holders, in different geometric configurations. © 2000 Optical Society of America

OCIS codes: 310.0310, 310.6870, 310.1860.

aa

1. Introduction

Correcting masks have been used in the past to ob-tain thickness uniformity in thin films prepared withphysical vapor deposition methods, such as in ther-mal evaporation, an electron-beam gun, and sputter-ing. The procedure for designing such correctingdiaphragms has been a combination of trial and errorand a semiempirical theory based on Knudsen’s lawsthat yields relationships for estimating the emissionproperties of vapor sources.1–6 A shortcoming ofthis method is that an experimental emission patterncannot be represented in a simply way with this mod-el,7 so the resulting correcting mask is not effectivefor achieving good uniformity, and some manual in-tervention is always needed to adjust its profile.

It is even possible to develop a correcting maskmerely by trial and error by depositing a sample filmwith a substrate holder under rotation and then mea-suring the thickness distribution, depending on thesymmetry of the deposition chamber and the shape ofthe fixtures. A first correcting mask is proposed,based on this experimental thickness distribution, byestimating its shape for obstructing more materialwhere the film is thicker and less where the film isthinner.8,9 This method has the disadvantage of be-ing very tedious because it takes at least 10 steps,depending on geometry, to obtain a solution thatguarantees ;3% of uniformity. Also the obtainedmask is useful for a given specific geometric configu-

The authors are with Centro de Investigaciones en Optica, Apar-tado Postal 1-948, C. P. 37000, Leon Gto., Mexico. F. Villa’se-mail is [email protected].

Received 27 July 1999; revised manuscript received 18 October1999.

0003-6935y00y101602-09$15.00y0© 2000 Optical Society of America

1602 APPLIED OPTICS y Vol. 39, No. 10 y 1 April 2000

ration, and the procedure for obtaining a correctingmask in a different geometry must be repeated toachieve the new profile.

When the method proposed in Ref. 7 is used torepresent the emission properties of vapor sourcesbased on experimental results, it is possible to deter-mine more precisely the shape of a correcting maskeven for different fixtures and geometric conditionsfor surface substrates.

2. Theory

It was proposed in Ref. 7 that the physical thicknessof a growing film at a given point in the substratesurface be expressed as

t 5 (j50

m

aj tj, (1)

where aj are coefficients determined from the exper-iment, tj 5 cos F cosj uyr92 ~for small sources com-pared with the distance to the substrate!, F is thengle between the normal to the substrate surfacend the radius vector r* 5 r 2 r1 joining a point on

the source with a point of the substrate surface asshown in Fig. 1. u is the angle between the normalto the point on the source and the vector r*, and m isthe number of terms of the series depending on thesource shape.

For extended sources Eq. ~1! becomes

tj~r1! 5 **P~ x, y!

vj~r, r1!u~r, r1!A~x, y!

ur 2 r1u j13 dxdy, (2)

where A~x, y! 5 @1 1 ~]zy]x!2 1 ~]zy]y!2#1y2 is thesource area function and P~x, y! is the projection ofthe source surface F~x, y, z! 5 0 on the x–y plane.The substrate surface is represented by S~x1, y1, z1! 5

ta

oadg

a

0 as shown in Fig. 1. We also call f 5 ¹Fyu¹Fu and s5 ¹Syu¹Su unitary vectors normal to the surface ofthe source and substrate, respectively. Finally v~r,r1! 5 f z ~r1 2 r! is the source function, and u~r, r1! 5s z ~r1 2 r! is the substrate function.

3. Correcting Masks for Rotating Substrates

It is well known also that the rotation of substratesimproves uniformity, and in some cases by usingplanetary systems it is possible to achieve a highdegree of film-thickness uniformity.10 The main dis-advantage with planetary systems is the complexityof choosing the correct geometry to attain the opti-mum uniformity and the difficulty of manufacturingcomplex mechanical fixtures. With simple rotatingsubstrates and correcting diaphragms it is possible toobtain great uniformity on large surfaces in a varietyof geometric configurations. One disadvantage ofcorrecting masks is material loss, which can be min-imized only by optimizing the geometric configura-tion.

According to Fig. 1, points on the substrate surfaceare given by the coordinates ~x1, y1, z1!, with therotation axis for the substrate holder aligned alongany of these coordinates. It is convenient to expressthe remaining pair of coordinates in polar form:w1 5 r cos c, w2 5 r sin c, where w1 and w2 are anycoordinate, x1, y1, or z1.

In the case of a rotating substrate we can expresshe thickness in Eq. ~2! as a function of polar vari-bles:

t#j@r1~r!# 5 *0

2p

tj@r1~r, c!#dc. (3)

Equation ~3! represents the accumulated thicknessf the film at a specific radius r and is valid only if thessumption is made that the emission is constanturing one rotation period and that the thicknessrown during a fraction of a period is negligible.In general, t#j@r1~r!# varies from point to point at the

substrate surface; then the purpose of a correcting

Fig. 1. General geometric configuration of a source–substratesystem. The surfaces of source and substrate are represented byF~x, y, z! and S~x1, y1, z1!, respectively.

mask is to obstruct selectively the material arrivingat the substrate to make t#j@r1~r!# a constant. In sucha way, one condition for defining the form of the maskis given by

t#j@r1~r!# 5 *w~r!1a

p1a

tj@r1~r, c!#dc 5 tj0, (4)

where w~r! defines the angular aperture of the mask,[ @0, py2# defines the orientation of the symmetry

axis of the mask, and tj0 is the minimum of t#j@r1~r!#with respect to r. In this way, once the emission lawof a given vapor source and the geometric configura-tion are known, it is possible to determine the profileof a correcting mask.

Equation ~4! can be applied only to sources thathave a maximum of two symmetry axes, e.g., theellipsoidal case. For spherical or disk sources, theorientation of the mask is independent of a.

For plane and spherical substrate holders, a sin-gularity may be found on the surface in the region ofinterception of the rotation axis, making it more dif-ficult to obtain good thickness uniformity despite thepresence of the mask. It is proposed in Ref. 5 thatthis problem can be solved with correcting mesh.However, some instability in uniformity can be found.In this case it is recommended that the central regionbe avoided as far as possible within a small radius~;1 cm!.

4. Projection Masks

In some geometric configurations, e.g., conical orspherical, it is difficult to fit exactly the curvature ofthe substrate holders with the correcting mask, souse of a projection diaphragm is recommended.When the source is small compared with the distancefrom the source to the substrate, we can think of anhorizontal plane mask positioned at a height zm 5hm. The relationships that define the projectionmask profile are given by

xm 5x1 2 xz1 2 z

~zm 2 z! 1 x, (5)

ym 5y1 2 yz1 2 z

~zm 2 z! 1 y. (6)

Here ~x, y, z! represents the position of the source,and ~x1, y1, z1! represents points on the substrate,particularly those denoting the uniformity mask pro-file, defined on its surface or close to it.

5. Typical Geometric Configurations

In the following examples we consider for the sub-strates, flat disk, cone, and spherical shell, the ellip-soidal convex source with an emission lobe describedby the parameters h0 5 0.5, a 5 0.25, b 5 1.5, and c 50.75 cm ~Ref. 7! and the emission law t 5 142.5t0 210.3t1 1 2500t5 2 2577t7. The same kind of sourceapplies with parameters h0 5 0, a 5 2.2, b 5 2.2, andc 5 1.1 cm and the emission law t 5 8.62t0 1 46.6t1 145.7t3 for cylinder substrates. These sources repre-

1 April 2000 y Vol. 39, No. 10 y APPLIED OPTICS 1603

m

dsdactcs

As

t

t

1

sent the most general configuration, and they werederived from experimental results.

Both sources are described by the surface function,

z~x, y! 5 cS1 2x2

a2 2y2

b2D1y2

2 h0, (7)

and the area function,

A~x, y! 5 S1 1 c2 x2ya4 1 y2yb4

1 2 x2ya2 2 y2yb2D1y2

. (8)

In all the cases we consider below the correctingasks are shadowed in each case.It is well known that the microstructure of thin films

epends on the angle of arrival F of molecules to theurface of the substrate.11,12 This is desirable in theesign of chambers and fixtures to keep the angle ofrrival of molecules as close as possible to normal in-idence to obtain thin films with good physical proper-ies. Among the different configurations that weonsider below, the only one that has a limited usefulurface due to this factor is the vertical cylinder.

A. Plane Substrate ~Horizontal Disk!

Unless otherwise stated, the source is always placedat the origin of the reference system, x 5 0, y 5 0.

gain, according to Ref. 7 and Fig. 2, the plane sub-trate and the source functions will be, respectively,

u~x, y! 5 z1 2 z~x, y!, (9)

v~x, y, r, c! 51

D~x, y! Hxx 2 r cos c

a2 1 yy 2 r sin c

b2

1 @z~x, y! 1 h0#z~x, y! 2 z1

c2 J , (10)

D~x, y! 5 Hx2

a4 1y2

b4 1@z~x, y! 1 h0#

c4 J1y2

. (11)

Fig. 2. Plane horizontal disk substrate: hm, is the height wherehe projection mask is placed.


Note that in Eqs. ~9!–~11! and in those that followwe have stated for clarity the explicit dependence offunctions on variables x, y, r, and c instead of r andr1.

Rotation around the z1 axis means that x1 5 r cosc and y1 5 r sin c, so Eq. ~2! can be expressed in thiscase as

tj~r, c! 5 *2b2

b

*2f ~ y!

f ~ y! vj~x, y, r, c!u~x, y!A~x, y!

ur 2 r1u j13 dxdy,

(12)

where f ~y! 5 a~1 2 y2yb2!1y2 and

ur 2 r1u 5 $~x 2 r cos c!2 1 ~y 2 r sin c!2

1 @z~x, y! 2 z1#2%1y2. (13)

The thickness distribution for radial directions x~solid curve! and y ~dotted curve! is shown in Fig. 3and its corresponding correcting mask on the surfaceof the substrate in Fig. 4 ~solid curve!. The projec-ion flat mask placed at hm 5 20 cm is shown in the

same figure ~dashed curve!.

Fig. 3. Thickness normalized distribution in directions x ~solidcurve! and y ~dashed curve! of the flat disk. In this case hp 5 32cm.

Fig. 4. Correcting ~solid curve! and projection ~dashed curve,hm 5 20 cm! mask for a flat disk.

a

Ts

B. Vertical Cylindrical Substrate

In this case the substrate to be coated is the curvedwall of a cylinder ~Fig. 5!.

The substrate function is given by

u~x, y, c! 5 cos c~R cos c 2 x! 1 sin c~R sin c 2 y!

(14)

nd the source function

v~x, y, c, z1! 51

D~x, y! Hxx 2 R cos c

a2 1 yy 2 R sin c

b2

1 @z~x, y! 1 h0#z~x, y! 2 z1

c2 J . (15)

In this case the rotation axis is z1 and the thicknessdistribution depends on the height of the cylinder, sot#j 5 t#j~z1! and

ur 2 r1u 5 @~x 2 R cos c!2 1 ~y 2 R sin c!2

1 @z~x, y! 2 z1#2#1y2. (16)

The thickness distribution for this curved surfaceis shown in Fig. 6, and its corresponding correctingmask extended on a plane is given in Fig. 7. In thiscase a projection mask is not practical, given thegeometry of the system.

Fig. 5. Vertical cylinder substrate of radius R and height hp.

Fig. 6. Normalized thickness distribution on the curved wall of avertical cylinder.

C. Spherical Shell Substrate Holder

This surface is shown in Fig. 8, and its correspondingsubstrate function is

u~x, y, r, c! 51rs

$r cos c~r cos c 2 x!

1 r sin c~r sin c 2 y!

1 @z1~r! 2 k#@z1~r! 2 z~x, y!#%. (17)

The source function and ur 2 r1u are given by Eqs.~10! and ~13!, respectively.

The thickness distribution for this spherical shell isshown in Fig. 9, and their corresponding correctingmask is seen from a top view in Fig. 10 ~solid curve!.

he projection flat mask placed at hm 5 15 cm is alsohown in Fig. 10 ~dotted curve!.

D. Cone Truncated Substrate

The parameters of this surface are given in Fig. 11,and its substrate function is

Fig. 7. Correcting mask on surface.

Fig. 8. Spherical dome-shaped substrate of radius rs.


f

r

1

where p 5 cot2 u, u 5 arcsin~h1yc1!, q 5 r0 cot u 1 hcot2 u, and

z1~r! 5 h 2 ~r 2 r0!tan u. (19)

As with the case of a spherical shell, the sourceunction and the corresponding expression for ur 2 r1u

are given by Eqs. ~10! and ~13!, respectively.The thickness distribution for this surface is shown

in Fig. 12, and the corresponding correcting mask isseen from a top view in Fig. 13 ~solid curve!. Theprojection flat mask placed at hm 5 15 cm is shown bya dashed curve.

Fig. 9. Normalized thickness distribution along the radius indirections x ~solid curve! and y ~dashed curve!. In this case nor-malization is relative to the maximum thickness in the x direction.The parameters of the dome are rs 5 28, k 5 4, h 5 32 cm.

Fig. 10. Correcting masks on the surface ~solid curve! and pro-jected on a plane with hm 5 15 cm ~dotted curve!.

u~r, r1! 5r cos c~r cos c 2 x! 1 r sin c

$~r cos c!2 1 ~


Fig. 11. Truncated cone substrate.

Fig. 12. Normalized thickness distribution along the radius inthe x ~solid curve! and the y ~dashed curve! directions. The pa-ameters of conical fixture are r0 5 5, h1 5 5, h 5 30, c1 5 28 cm.

~r sin c 2 y! 1 @q 2 pz1~r!#@z1~r! 2 z~x, y!#

r sin c!2 1 @q 2 pz1~r!#2%1y2 , (18)

w

s

rIi

~

E. Horizontal Cylinder Substrate

In this case we have a cylinder of radius rc ~Fig. 14!ith a substrate function:

u~x, y, c! 51rc

$rc cos c~rc cos c 2 x!

1 @z1~c! 2 k#@z1~c! 2 z~x, y!#%. (20)

Fig. 14. Horizontal cylinder substrate of radius rc ~drum vacuumchamber!.

The rotation axis is now defined by the line y1 5 k,so that z1~c! 5 rc sin c 1 k and x1 5 rc cos c. Theource function will be

v~x, y, y1, c! 51

D~x, y! Hxx 2 rc cos c

a2 1 yy 2 y1

b2

1 @z~x, y! 1 h0#z~x, y! 2 z1~c!

c2 J ,

(21)

ur 2 r1u 5 $~x 2 rc cos c!2 1 ~y 2 y1!2

1 @z~x, y! 2 z1~c!#2%1y2. (22)

For this configuration the distribution along the xand the y directions is shown in Fig. 15, and its cor-esponding mask is extended in a plane in Fig. 16.n this case also a projection mask is not of practicalnterest.

6. Correcting Mask for Coevaporation

The coevaporation of two materials is also of practicalinterest, although there are some physical limita-tions regarding the uniformity in the refractive indexof the resulting composite thin film along the sub-

Fig. 15. Normalized thickness distribution along the circle y 5 0dotted curve! and along a line x 5 0 ~solid curve! on the surface of

the cylinder ~rc 5 22; length, 64; k 5 28 cm!.

Fig. 16. Correcting mask on surface.

Fig. 13. Correcting masks on the surface ~solid curve! and pro-jected on a plane with hm 5 15 cm.


0

y

u

d

x

c

1

strate surface. Let us analyze the case of two vaporsources with different materials and different emis-sion patterns emitting simultaneously at constantrates. For simplicity we assume that there is nochemical reactivity between evaporating materials.In practice, if we are growing a composite thin film, itis necessary to consider the effective medium theorythat gives an approximation to determine the effec-tive refractive index of a mixture composed com-monly with the phases of constituent materials andvoids. The study of this problem is beyond the scopeof this paper, so we assume that the accumulatedthickness on the substrate surface is proportional tothe rates of deposition of each material individually.This is equivalent to assuming that we have no voidsin the mixture.

In these conditions let us consider the geometricconfiguration in Fig. 17 with a plane disk substrate of22-cm radius and z1 5 32.5 cm. Two ellipsoidal con-vex sources are placed at points ~x0, 0, 0! and ~0, y0,0!. The lengths of the axes of the sources are a1 52.2, b1 5 2.2, c1 5 1.1 cm and a2 5 2, b2 5 1.1, c2 5.55 cm. For both sources we have h0 5 0 and the

same emission law:

t 5 2379.47t0 1 1220t1 2 904.94t3. (23)

Normalized thickness distributions for a rotatingsubstrate as a function of radius are shown in Fig. 18~solid curve! for each source independently when x0 5

0 5 20 cm.Resultant emission pattern with both sources emit-

ting simultaneously depends on deposition rates ofeach material. We can define the total accumulatedthickness with

t~r! 5 v1 t1~r! 1 v2 t2~r!, (24)

where v1, v2 are the maximum rates of depositionachieved with each source at points of the radius ofthe rotating substrate where t1~r! and t2~r! have theirrespective maximum. In Fig. 19 we have the nor-malized thickness distribution ~solid curve! when the

Fig. 17. Geometric configuration with a double source and a planerotating substrate disk.


rates of evaporation are v1 5 1 and v2 5 5 ~arbitrarynits! and the sources are placed at x0 5 y0 5 20 cm.

A quite different thickness distribution results if therates of deposition are v1 5 1 and v2 5 1 ~Fig. 19,

ashed curve!. In Fig. 18 ~dashed curve! we showthe thickness distribution of the same sources at po-sitions given by x0 5 y0 5 10 cm, and in Fig. 19~dash–dot curve! we have the distribution producedby both sources under codeposition with v1 5 1 andv2 5 5.

Corresponding correcting masks for two cases, ~a!

0 5 y0 5 10; ~b! x0 5 y0 5 20 cm, both with rates ofv1 5 1, v2 5 5, are shown in Fig. 20.

7. Efficiency of Masks

A correcting mask obstructs the arrival of part of thematerial emitted by the source, and the loss of mate-rial is independent of the angular orientation of themask. Efficiency h is defined as the quantity of ma-terial that is actually arriving at the rotating sub-strate with the correcting diaphragm, divided by the

Fig. 18. Normalized thickness distributions for sources placed inthe xy plane at points ~x0, 0! and ~0, y0!; Case ~a! x0 5 y0 5 20 ~solidurve!; case ~b! x0 5 y0 5 10 cm ~dashed curve!.

Fig. 19. Normalized thickness distribution for coevaporated ma-terials with rates of deposition v1 5 1 and v2 5 5: Case ~a! solidcurve; case ~b! dashed curve. Case ~a! with v1 5 1 and v2 5 1~dash–dot curve!.

t

ramaha

n

l0tecit

c

c

total material arriving to the rotating substrate sur-face without the mask,

h 51p (

i50

p H*w~r!1a

p1a

@r1~ri, c!#dcY *0

p

tj@r1~ri, c!#dcJ ,

(25)

where p is the number of points it takes to calculatehe integrals.

Figure 21 and 22 show efficiency as a function ofadius or height for the different cases consideredbove. In all these cases we consider correctingasks with their symmetry axis aligned with the x

xis. The case of a mask for a spherical substrateolder, with its symmetry axis aligned with the yxis, is shown in Fig. 10 ~dash–dot curve!. The ef-

ficiency of this mask is the same as that aligned withthe x axis.

8. Real Vapor Sources and Their Correcting Masks

To show how the method works with real vaporsources, in Fig. 23 we have the thickness distribution

Fig. 20. Correcting masks. Case ~a! solid curve; case ~b! dashedurve.

Fig. 21. Efficiency as a function of radius for plane ~solid curve!,one ~dotted curve!, and spherical ~dash–dot curve! substrates.

for the x ~sphere symbols! and the y directions ~tri-angles! produced by a rectangular source of 0.5 cm 32.2 cm with titanium monoxide evaporated reac-tively. One virtual emission surface that fits suchdistribution is an ellipsoidal convex of dimensionsa 5 b 5 2.2 cm, c 5 1.1 cm with an emission law givenby t 5 2379.47t0 1 1220t1 2 904.94t3. Note thesymmetry of emission lobe around the z axis. Theuniformity mask corresponding to this configurationis shown in Fig. 24, and the corresponding thicknessdistribution obtained experimentally with this maskis shown in Fig. 23 ~diamond symbols!. The thick-

ess uniformity is achieved within a precision of 3%.In the examples given above, the uniformity calcu-

ated theoretically under any configuration is within.1% or less. However, experimental limiting fac-ors such as variations in the emission lobe duringvaporation due to redistribution of material, the pre-ision on positioning the mask, and the intrinsic lim-tations of our method of measuring the thickness ofest film, among others, represent a big deal for

Fig. 22. Efficiency as a function of length for vertical ~solid curve!and horizontal cylinders ~dotted curve!.

Fig. 23. Experimental normalized thickness distribution in twoperpendicular directions along the x axis ~spheres! and y axis~triangles!. Experimental normalized thickness distribution onthe plane disk with the correcting mask ~diamonds!.


1

achieving better uniformity than that obtained ex-perimentally.

9. Conclusions

We have shown that in almost every standard geo-metric configuration for real extended sources andsubstrates it is possible to determine the diaphragmprofile in a simple way by using vectorial formalism.As mentioned above, once we determined the emis-sion law of a vapor source, with this method it ispossible to design correcting masks in a variety ofconfigurations by performing only a few experiments.

In future research we will consider the effect of

Fig. 24. Correcting mask for a real source.


errors due to large extended sources and rotatingmasks.

We thank Jesus Nieto and Octavio Pompa for help-ful experimental support.

References1. H. K. Pulker, Coatings on Glass ~Elsevier, Amsterdam, 1984!.2. A. I. Usoskin, “Correcting diaphragms for improving the thick-

ness uniformity of vacuum coatings,” Sov. J. Opt. Technol. 51,471–474 ~1984!.

3. Zh. P. Trofimova, V. M. Kholodov, T. I. Demidovich, and Ya. V.Petlitskaya, “Analysis of the condensate distribution and se-lection of correction masks for obtaining uniform thicknessoptical coatings,” Sov. J. Opt. Technol. 54, 357–359 ~1987!.

4. G. V. Panteleev, A. A. Zhuravlev, and V. V. Morshakov, “Cor-recting mask for increasing the thickness uniformity of vac-uum coatings,” Sov. J. Opt. Technol. 55, 547–548 ~1988!.

5. C. Grezes-Besset, R. Richier, and E. Pelletier, “Layer unifor-mity obtained by vacuum evaporation: application to Fabry–Perot filters,” Appl. Opt. 28, 2960–2964 ~1989!.

6. H. A. Macleod, Thin Film Optical Filters ~McGraw-Hill, NewYork, 1989!.

7. F. Villa and O. Pompa, “Emission pattern of real vapor sourcesin high vacuum: an overview,” Appl. Opt. 38, 695–703 ~1999!.

8. K. H. Behrndt, “Film-thickness and deposition-rate monitoringdevices and techniques for producing thin films of uniformthickness,” Phys. Thin Films 3, 1–59 ~1966!.

9. R. R. Willey, Practical Design and Production of Optical ThinFilms ~Marcel Dekker, New York, 1996!.

10. A. Perrin and J. P. Gailliard, “Planetary system for high uni-formity deposited layers on large substrates,” in Thin Films forOptical Systems, K. H. Guenther, ed., Proc. SPIE 1782, 238–244 ~1992!.

11. L. Holland, Vacuum Deposition of Thin Films ~Chapman &Hall, London, 1970!.

12. A. Musset and I. Stevenson, “Thickness distribution of evap-orated films,” in Optical Thin Films and Applications, H. Herr-mann, ed., Proc. SPIE 1270, 287–291 ~1990!.

correction masks for thickness uniformity in large-area thin films

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