Symmetric magnetic doublet for charged particle beam lithography

Symmetric magnetic doublets are disclosed that image a reticle onto a sensitized substrate using a charged-particle beam. The symmetric magnetic doublet comprises an object-side lens and an image-side lens and satisfies certain quantitative conditions. If the object-side lens has a length S.sub.1 and pole-piece apertures of radii R.sub.1, R.sub.2, and the image-side lens has a length S.sub.2 and pole-piece apertures of radii R.sub.3, R.sub.4, then a lens according to an embodiment of the invention produces a demagnification of 1/m in an object-image distance L between the reticle and the substrate. This embodiment satisfies the relations R.sub.3 =R.sub.2 /m, R.sub.4 =R.sub.1 /m, S.sub.2 =S.sub.1 /m, and ##EQU1##

FIELD OF THE INVENTION 
The invention pertains to magnetic lenses for charged-particle-beam 
lithography. 
BACKGROUND OF THE INVENTION 
Charged-particle-beam pattern-transfer apparatus use a charged-particle 
beam, such as an electron beam, to project a pattern from a reticle or 
mask onto a wafer. In such apparatus, a rotationally-symmetric magnetic 
lens typically images the reticle onto the wafer. This lens must produce 
clear, undistorted images. One satisfactory lens type is a magnetic 
doublet having two magnetic lenses that produce opposite magnetic fields. 
The trajectory of a charged particle in a charged-particle optical system 
is usually described in terms of a paraxial trajectory and aberrations. In 
a rotationally symmetric electric or magnetic field, the trajectory of a 
charged particle is calculated based on the coordinates (x.sub.0, y.sub.0) 
and propagation angles (.alpha..sub.1, .alpha..sub.2) of the particle at 
the wafer plane. The aberrations of the optical system are expressed in 
terms of polynomials of order 2n+1, where n.gtoreq.1. Terms of order 2n+1 
represent the (2n+1)th order aberrations. 
While most aberrations are represented by terms of order 3 or higher, 
chromatic aberration effects even the paraxial trajectory. Chromatic 
aberration is caused by the spread of charged particle energies in the 
charged particle beam and chromatic aberration therefore occurs at orders 
of (2n-1) or larger, where n.gtoreq.1. 
If the charged particle trajectory is expressed in terms of the complex 
coordinates w=x+iy, the total third-order geometric aberration as a 
function of the complex coordinate .beta.=x.sub.0 +iy.sub.0 and the 
complex propagation angle .alpha.=.alpha..sub.1 +i.alpha..sub.2 at the 
wafer is given by: 
##EQU2## 
wherein .alpha.* and .beta.* are the complex conjugates of .alpha. and 
.beta. respectively, V is the accelerating voltage and .DELTA.V is the 
charged-particle-beam energy spread. The various aberration coefficients 
are K.sub.sph (spherical aberration), K.sub.coma-l (longitudinal coma), 
K.sub.coma-r (transverse or radial coma), K.sub.fc (field curvature), 
K.sub.astig (astigmatism), K.sub.dis (distortion), K.sub.t-chro 
(transverse chromatic aberration), and K.sub.a-chro (axial chromatic 
aberration). 
In a magnetic doublet, distortion, chromatic aberration, and image rotation 
generated by a first lens offset distortion, chromatic aberration, and 
image rotation produced by a second lens. For this reason, the transverse 
chromatic aberration coefficient K.sub.t-chro and the distortion 
coefficient K.sub.dis are zero. However, the magnetic doublet exhibits 
other types of aberrations that depend upon fields produced by the lens. 
In charged-particle-beam pattern-transfer apparatus, image blur caused by 
Coulomb interactions of the charged particles in the beam limits 
throughput. In order to reduce this image blur, the beam current density 
is reduced. One method of decreasing current density without decreasing 
throughput is to irradiate a large area of the reticle with a large 
diameter beam. Increasing the beam numerical aperture also reduces image 
blur due to Coulomb interactions. 
The aberrations of the magnetic doublet are not determined solely by the 
aberrations of the individual lenses of the doublet, and optimization of 
the individual lenses does not ensure optimization of the doublet. In 
addition, the actual aberrations realized during use depend upon the 
propagation direction of the beam and location of the beam on the reticle. 
For example, if the reticle is demagnified with a demagnification 1/m onto 
the wafer, then the initial coordinate of the beam is m.beta. at the 
reticle. If the initial coordinate of the beam is increased (i.e., the 
beam propagates farther off-axis), then the aberrations are changed 
according to Equation 1. For this reason, lens-system design calculations 
are complex and it is difficult to optimize such lenses. 
Therefore, it is an object of the invention to reduce the blur associated 
with geometric aberrations and Coulomb interactions. 
SUMMARY OF THE INVENTION 
In one embodiment of the invention, a magnetic lens system for forming an 
image of an object irradiated by a charged-particle beam is provided. The 
object is typically a reticle and the image of the reticle is generally 
formed on a surface of a sensitized substrate, such as a semiconductor 
wafer coated with a resist that is sensitive to a charged-particle beam. 
The magnetic lens system comprises an object-side lens system of length 
S.sub.1 and having an object-side aperture of radius R.sub.1 and an 
image-side aperture of radius R.sub.2, and an image-side lens system of 
length S.sub.2 having an object-side aperture of radius R.sub.3 and an 
image-side aperture of radius R.sub.4. The image is demagnified by 1/m and 
the magnetic lens system satisfies the doublet symmetry conditions: 
EQU R.sub.3 =R.sub.2 /m, 
EQU R.sub.4 =R.sub.1 /m, 
EQU S.sub.2 =S.sub.1 /m, 
and 
EQU .vertline.z.sub.2 -z.sub.i .vertline.m=.vertline.z.sub.1 -z.sub.o 
.vertline., 
wherein the object and the image are positioned along an axis at 
coordinates z.sub.o and z.sub.i, respectively, the object-side lens system 
and the image-side lens system are positioned along the axis at 
coordinates z.sub.1 and Z.sub.2, respectively, and L is the distance 
between the image and the object and is given by L=.vertline.z.sub.i 
-z.sub.o .vertline.. Because the lens satisfies the doublet symmetry 
conditions, it is referred to as a symmetric magnetic doublet. 
A magnetic lens according to an embodiment of the invention satisfies an 
additional condition: 
##EQU3## 
Another embodiment comprises a magnetic lens system that satisfies a 
further additional condition: 
##EQU4## 
In a further embodiment of the invention, the distance L=400 mm. 
Yet another embodiment of the invention comprises a magnetic lens system 
satisfying the doublet symmetry conditions and the condition: 
##EQU5## 
wherein L=400 mm is the distance between the object and the image 
Still another embodiment of the invention comprises a magnetic lens system 
comprising an object-side lens system of length S.sub.1 and having an 
object-side aperture of radius R.sub.1 and an image-side aperture of 
radius R.sub.2, and an image-side lens system of length S.sub.2 having an 
object-side aperture of radius R.sub.3 and an image-side aperture of 
radius R.sub.4. The image-side lens system is displaced a distance 
.DELTA.z toward the object-side lens system from the doublet-symmetry 
conditions that are satisfied if .DELTA.z=0: 
EQU R.sub.3 =R.sub.2 /m, 
EQU R.sub.4 =R.sub.1 /m, 
EQU S.sub.2 =S.sub.1 /m, 
and 
EQU .vertline.z.sub.2 -z.sub.i .vertline.m=.vertline.z.sub.1 -z.sub.o 
.vertline.. 
In another embodiment of the invention, the displacement .DELTA.z satisfies 
the condition: 
##EQU6## 
Additional embodiments are provided in which the image-side lens system is 
displaced toward the object-side lens system by the distance .DELTA.z and 
additional conditions are satisfied. In one of these embodiments, the 
distance L between the image and the object satisfies the condition: 
##EQU7## 
Alternatively, the magnetic lens satisfies the condition: 
##EQU8## 
In an example embodiment of the invention, the length S.sub.1 =300 mm, 
R.sub.1 =R.sub.2 =40 mm, and L=400 mm. 
The foregoing and other features and advantages of the invention will 
become more apparent from the following detailed description of a 
preferred embodiment which proceeds with reference to the accompanying 
drawings.

DETAILED DESCRIPTION 
With reference to FIG. 1, a reticle 1 or other object is irradiated by an 
electron beam EB produced by an electron gun EG. The electron beam EB 
propagates along a z-axis 10 to a sensitized substrate 2 or other image 
plane. The sensitized substrate 2 is typically a semiconductor wafer 
coated with an electron-beam-sensitive resist. A symmetric magnetic 
doublet 20 comprising an object-side lens 3 and an image-side lens 4 
images the reticle 1 onto the substrate 2 with a demagnification of 1/m. 
For convenience, locations are referred to as on an object side or an 
image side if closer to the reticle 1 or the sensitized substrate 2, 
respectively, as measured along the z-axis 10. 
The reticle 1 and the substrate 2 are positioned along the z-axis at 
coordinates z.sub.0 and z.sub.i, respectively, and are separated from each 
other by a distance L=.vertline.z.sub.i -z.sub.o .vertline.. The 
object-side lens 3 and the image-side lens 4 are located along the z-axis 
10 at coordinates z.sub.1 and Z.sub.2 and have lengths S.sub.1 and 
S.sub.2, respectively. A pole piece 5 of the object-side lens 3 has an 
object-side central aperture 12 of radius R.sub.1 and an image-side 
central aperture 14 of radius R.sub.2, respectively; the image-side lens 4 
has a pole piece 7 having object side and image side apertures 16, 18 of 
radii R.sub.3, R.sub.4, respectively. 
The lens 20 substantially satisfies the following conditions that are 
referred to herein as the "doublet-symmetry" conditions: 
EQU R.sub.3 =R.sub.1 /m, 
EQU R.sub.4 =R.sub.2 /m, 
EQU S.sub.2 =S.sub.1 /m, 
and 
EQU .vertline.z.sub.2 -z.sub.i .vertline.m=.vertline.z.sub.1 -z.sub.o 
.vertline.. 
A lens substantially satisfying these conditions is referred to as a 
"symmetric magnetic doublet." 
For convenience, the performance of the lens 20 (satisfying the symmetric 
magnetic-doublet conditions) is described with reference to illustrative 
parametric values for the dimensions of the lenses 3, 4 and the properties 
of the electron beam EB. The illustrative values are the propagation angle 
.alpha.=8 mrad, the coordinate .beta.=0.5 mm, the accelerating voltage 
V=100 kV, the beam energy spread of 5 eV, and the object-image distance 
L=400 mm. The propagation angle .alpha. and the coordinate .beta. are 
measured at the image plane, i.e., at the sensitized substrate 2. 
The symmetric magnetic doublet 20 is first evaluated at specified values of 
the lens dimensions. For purposes of illustration, the length S.sub.1 =300 
mm and the radii R.sub.1, R.sub.2 of the pole piece 5 are R.sub.1 =R.sub.2 
=40 mm. The total image blur produced by the lens 20 is estimated as 0.12 
.mu.m based on the square root of the sum of the squares of the 
third-order aberrations. 
The parameters R.sub.1, R.sub.2, and S.sub.1 can be varied while satisfying 
the doublet symmetry conditions. E.g., with reference to FIG. 2, the 
calculated image blur is graphed as a function of R.sub.1 for S.sub.1 =300 
mm and R.sub.2 =40 mm. The smallest image blur is about 0.12 .mu.m and is 
obtained for 20 mm.ltoreq.R.sub.1 .ltoreq.40 mm. For an image blur of this 
magnitude, the allowable image-blur tolerance is about 10% of the minimum 
image blur, so that the image blur is acceptable for 20 mm.ltoreq.R.sub.1 
.ltoreq.100 mm. The image blur is acceptably small for R.sub.1 at least as 
small as 10 mm as is shown in FIG. 2. However, manufacture of the pole 
piece 7 becomes difficult if R.sub.1 is small, so a minimum value of 20 mm 
for R.sub.1 is selected to enable simple manufacture. 
With reference to FIG. 3, image blur is graphed as a function of R.sub.2 
for S.sub.1 =300 mm and R.sub.1 =40 mm. The smallest value of calculated 
image blur is 0.12 .mu.m and is obtained at R.sub.1 =40 mm. Allowing a 10% 
tolerance in the image blur, the acceptable range for R.sub.2 is 20 
mm.ltoreq.R.sub.2 .ltoreq.100 mm. While the ranges for R.sub.1 and R.sub.2 
obtained from FIGS. 2-3 are calculated for L=400 mm, small variations in L 
do not change the results. 
The radii R.sub.1, R.sub.2 can be selected almost independently of the 
object-image distance L. Therefore, aberrations are reduced to acceptable 
levels with 20 mm.ltoreq.R.sub.1, R.sub.2 .ltoreq.100 mm for a wide range 
of values of the image-object distance L. 
With reference to FIG. 4, image blur is graphed as a function of the length 
S.sub.1 of the lens 3 with R.sub.1 =R.sub.2 =40 mm. Although the blur is 
calculated for distances S.sub.1 as large as S=360 mm, because the 
object-image distance L is fixed at 400 mm, the length S.sub.1 must be 
less than 320 mm for an actual lens. The blur is minimized at S.sub.1 =300 
mm with an approximate optimum range of 290 mm.ltoreq.S.sub.1 .ltoreq.320 
mm. In terms of the distance L and the magnification m, the distance 
S.sub.1 satisfies the following condition: 
##EQU9## 
In these calculations, the lengths S.sub.1, S.sub.2 and the 
pole-piece-aperture radii R.sub.1, R.sub.2, R.sub.3, R.sub.4 are varied so 
that the doublet-symmetry conditions remain satisfied. 
Image blurs have also been determined for configurations of the lenses 3, 4 
that initially satisfy the doublet-symmetry conditions but in which the 
lens 4 is moved toward the lens 3 so that doublet-symmetry conditions are 
no longer satisfied. With reference to FIG. 5, the image blur is graphed 
as a function of the position Z.sub.2 of the lens 4 for S.sub.1 =300 mm 
and R.sub.1 =R.sub.2 =40 mm. Values of Z.sub.2 in the range 359-360 mm 
correspond to the location of the lens 4 satisfying double symmetric 
magnetic conditions. The image blur has a minimum value of about 0.108 
.mu.m for 356 mm.ltoreq.Z.sub.2 .ltoreq.357 mm. This range can be 
expressed as a displacement .DELTA.Z.sub.2 along the z-axis 10 from an 
initial position in which the doublet-symmetry conditions are satisfied. 
The displacement .DELTA.Z.sub.2 of the lens 4 toward the lens 3 is given 
by: 
##EQU10## 
wherein the doublet-symmetry conditions are satisfied if the displacement 
.DELTA.Z.sub.2 =0. 
The blurs of FIGS. 2-5 are calculated for a fixed propagation angle .alpha. 
and coordinate .beta. at the sensitized substrate 2. The blurs result 
mainly from third-order geometric aberrations. FIGS. 6-7 display image 
blur as a function of the propagation angle .alpha. and the coordinate 
.beta., respectively. In FIG. 6, the image blur for the coordinate 
.beta.=0.5 mm is graphed as a function of the propagation angle .alpha.. 
In FIG. 7, the image blur for a propagation angle .alpha.=8 mrad is 
graphed as a function of the coordinate .beta.. The blurs of FIGS. 6-7 are 
calculated for R.sub.1 =R.sub.2 =40 mm and S.sub.1 =300 mm, 240 mm, 360 
mm, corresponding to curves A, B, C, respectively. The image blur is 
smallest for S.sub.1 =300 mm and the minimum values of .alpha. and .beta.. 
In addition, the image blur is smaller for S.sub.1 =300 mm than S.sub.1 
=240 mm or S.sub.1 =360 mm for all values of the propagation angle .alpha. 
and coordinate .beta.. 
Geometric aberrations of the lenses 3, 4 cause the image blurs of FIGS. 
2-7. An additional aberration is caused by Coulomb interactions of the 
electrons in the electron beam EB. This aberration is conveniently 
calculated using a Monte Carlo simulation. The calculated total blur is 
then the square root of the sum of the squares of the blurs due to Coulomb 
interactions and the geometric aberrations. The calculations of the blur 
caused by Coulomb interactions of the electrons are illustrated with the 
lens parameters A of FIG. 6, a beam current of 25 .mu.A, and a 250-.mu.m 
square beam. FIG. 8 shows the calculated blur; the total blur is smallest 
at a propagation angle a of about 7 mrad and an acceptable range for the 
propagation angle .alpha. is 5.8 mrad.ltoreq..alpha..ltoreq.8.0 mrad. 
Lens systems for charged-particle beams are generally optimized by 
optimizing individual lenses. In contrast, the symmetric magnetic doublet 
is improved by optimizing the entire lens system, i.e., lenses 3, 4, 
simultaneously. 
The lenses 3, 4 are illustrated in FIG. 1 as individual magnetic lenses, 
but can comprise lens systems having one or more lenses and are referred 
to as lens systems. 
Having illustrated and demonstrated the principles of the invention in a 
preferred embodiment, it should be apparent to those skilled in the art 
that the preferred embodiment can be modified in arrangement and detail 
without departing from such principles. We claim as the invention all that 
comes within the scope of these claims.