- Email: [email protected]
Computation of multi-lens focusing systems
Nuclear Instruments and Methods in Physics Research A298 (1990) 45-55 North-Holland
45
Computation of multi-lens focusing systems Bohumila Lencovâ * and Michal Lenc
Institute of Scientific Instruments, Czechoslovak Academy of Sciences, Kr6lovopolskd 147, CS - 612 64 Brno, Czechoslovakia
A computational method based on the trajectory method is presented for the evaluation of systems of magnetic electron lenses. A consistent set of aberration coefficients is derived for third-order geometrical and first-order chromatic aberrations. The coefficients can be evaluated for an arbitrary aperture position, and also for a defocused image position due to the inclusion of the aberration of the gradient. A computer program called SYSTEM incorporating these new features has been written and the use of the program is illustrated with a suitable example.
1. Introduction In the computation of a system of electron lenses such as the imaging or illumination system of a transmission electron microscope, it is highly desirable to be able to evaluate the performance of the system as a whole. This is so because the aberrations of individual lenses combine to give the overall aberration of the system . Moreover, the beam is usually defined by the position and size of apertures, and the aberrations are influenced accordingly. The observation plane (screen or film) does not usually coincide with the Gaussian image plane of the system, and the aberrations, in particular the distortion, may differ in different reference planes. There are basically two methods in use for the computation of the aberrations of a system of magnetic electron lenses (for further details see ref. [1]) : a) The matrix method, based on a general perturbation theory, which evaluates for each lens excitation a matrix defining both the paraxial image properties (the focal length and the position of principal planes) and the third-order geometrical and first-order chromatic aberrations . For a system of lenses, these matrices are computed for each lens, and the system properties are determined by multiplying the relevant lens and transfer matrices in a defined way. This approach, pioneered in electron optics by Hawkes [2,3], was applied in a computer program by Maclachlan [4,5], but the inclusion of the effect of the aperture position and image defocus were not considered . These programs were restricted to model fields only, and have never been widely available. A disadvantage of the matrix method is that it is not very transparent, as the intermediate results are not easy to interpret. It is also necessary to mix real and asymptotic lens properties for the computation of transmission electron microscopes, and there can also be some problems in the case of overlapping focusing fields of individual magnetic lenses . On the other hand, so far only in the matrix method have the aberrations of slope been included . b) The trajectory method, based on the variation of parameters . Two independent trajectories begin at the object, and the image position is determined with the help of a trajectory which starts, with unit slope, on the optical axis. In order to obtain the off-axis aberrations, a second independent trajectory is used, starting at unit distance from the axis in the object plane . The second trajectory can be defined either to have a zero slope at the object plane or to cross the axis in the aperture plane. The aberrations in the image plane are then evaluated from integrals expressed with the help of the two independent trajectories . The trajectory method has the virtue of being straightforward, and the beam parameters in an arbitrary plane can be interpreted easily . It can also be extended to handle the aberrations for an arbitrary aperture position and in an arbitrary observation plane, not only in the Gaussian image plane, by including an * Present address: Particle Optics Group, Department of Applied Physics, TU Delft, Lorentzweg 1, 2628 CJ Delft, The Netherlands . 0168-9002/90/$03 .50 0 1990 - Elsevier Science Publishers B.V. (North-Holland)
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B Lencouli, M. Lenc / Multi-lens focusing systems
additional aberration coefficient related to the aberration of the gradient, as shown below. It is then possible to evaluate from one specific set of aberration coefficients, evaluated with the aid of the second independent trajectory starting at the object parallel to the axis (i.e. supposing the aperture plane to be at the zero point of this trajectory, also called the aberrations for a system without an aperture) the set of aberration coefficients for an arbitrary aperture plane position . The aberrations can be then also expressed at an arbitrary plane behind the magnetic lens system, which enables us to handle a defocused image, and at any plane between the lenses, to separate the effect of individual lenses. We have thus derived a consistent set of formulas for the computation of aberration integrals, which uses the same general expression for all isotropic aberrations including the aberration of the gradient, and another expression for anisotropic aberrations and for chromatic aberrations . Then we can easily evaluate the aberration coefficients for an arbitrary position of reference and aperture planes. The third-order aberrations of the gradient are usually omitted in the trajectory method, but they can also be easily expressed here, as shown below. The interrelation between the various aberration coefficients can thus easily be established . We have previously derived a suitable method for fast and accurate paraxial trajectory tracing and for an estimate of lens excitation for given focusing conditions, and implemented it together with aberration computations in a program called FOCUS [6,7] . For the design of a system of magnetic electron lenses a new program called SYSTEM has been written . By using the experience obtained with FOCUS, it was possible to check the performance of the new program . 2. Theory 2.1. Basic paraxial trajectories We define the paraxial performance of the electron optical system with the help of two independent trajectories ra (z) and rb(z), which obey the paraxial ray equation r " (Z)
+ K 2B2 (Z)r(z)
= 0,
with K = 2(e/2moV, )1/2 , e/m o being the ratio of electron charge to mass, and Vr is the relativistically corrected electron beam voltage V: Vr = V(1 + 0 .978 x 10 - 6 V) .
(2)
Electrons starting at the object in a plane containing the axis (the meridional plane) rotate with this plane by an angle O(Z) = K
f-B(z) dz. ,> Z
The two basic trajectories ra(z) and rb(z) can be conveniently defined by their initial coordinates in the object plane z = zo : ra( Z.) =0 ' rb(Z.) = 1 >
ra(Zo) -1, rb(zo) = 0.
The electron trajectory can also be expressed in the complex notation by its coordinates in the (x, y) plane as w(z) = x(z) + iy(z) = r(z) exp(iß(z)) either as W(Z) -
aowa(Z) +Nowb(Z) ,
(5)
or as W(Z) = ( a a +
Äßo)wa(Z)+ßoWb(Z) - aaWa(Z)+ßa Wc(Z)>
(6)
B . Lencoafi, M. Lenc / Multi-lens focusing systems
47
with X defined with the help of the trajectory coordinates in an aperture plane, z = za , thus X - - rb(Za)1ra( Za)' The new trajectory wJz), which is equal to zero in the aperture plane z = Z a , need not be calculated explicitly as it can easily be evaluated from the trajectories wa and wb. The aperture angle ao used here is the slope in the object plane only provided that the magnetic field there is equal to zero ; otherwise it is related to the generalized momentum in the rotating coordinate system ao =w ' (zo )-iKB(zo )ßo .
ßo is, as usual, the coordinate in the object plane, i.e. ßo = w(z o ) . In the aperture plane (rotated for convenience by an angle - B(za )) we have _ i6(Za) . ßa -a ra(Za)-w(Za)e_ 2.2. Third-order geometrical aberrations The deviation of the ray from the paraxial trajectory can be written with the help of aberration coefficients as AO(z) =Sa2âo + KLao,ß0âo + KRao0ß0 + Fa00 (10) ß ß0 + Aß02â0 + Dp0)2ß0, 2 or as (11) Ow(z) e- 'e(Z )=Sa2 o a +K Ra a2a ßo +Fa a a ßo ßo +A a ,(32â o a +Da lao o a aâa +KLaa a ßâ where S to D and Sa to Da are complex functions of the coordinate z, namely S=Sorb- (Ko - iko + )ra , Il
KL =2(Ko +iko)rb -(Fo +Ao +iKB(z o ))ra , KR = (Ko - iko)rb - (Ao - ia o - 2iKB(z o ))ra , F-- (F,, +A .)r b - 2(D,) - ldo + I K 2 B2 (zo)ra , A=(Ao +ik o )r b -(Do +ido -2 2B2 (zo))ra,
(12)
3 3(ZO) 1 2 B(zo)B i 3 ( ZO ) - -g1KB (z o ))ra . D= (Do+1d,))rb- (Eo + ZK (zo) + '2 1K B The functions So, Ko, Fo , Do , Eo and k o , a o and do are real functions, and for z = z,, where ra(z,) = 0 and rb(z,) = M gives the magnification M, they are the usual coefficients of the third-order aberrations related to the object plane. The additional function Eo in eq. (12) is related to the aberrations of the gradient, and thus it has no influence on the aberration in the image plane. This additional coefficient is normally not included in the treatment of the trajectory method, but it is vital for the possibility of calculating the aberrations in a plane other than the image plane, and for the evaluation of the aberrations of gradient. The functions So to do may be evaluated with the help of formulas :
So =11(a,a,a,a), Ko=I,(a,b,a,a), Fo=I,(a,b,a,b) + Ao=I,(a,a,b,b),
D. I, (a,b,b,b), Ea = I,(b,b,b,b), ko = j2(a,a), a o =2I2(a,b),
~~
foZ K2B2(z) dz, Z
(13)
do = l2(b,b),
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48
where we have denoted Z h(a,ß,y,8) = z f llrarp - K2BZrarp)l rv rs - K2BZrvrs) zo +KZ[ B~ra rp +B(ra rß+rârß ),[B ' r,rs +B(r,rs+rYrs )~} dz,
,24, P)
= 2K fo [B(rarß
+ KZBZrarp) + -1B ' (r«rß + ra rß) - 4 dz ( B ' rarR~] dz .
(14) (15)
The symmetry properties of the integral Il imply that So , F,, and Eo are always positive . The form chosen for the aberration integrals is suitable for further numerical computation. If nonconsistent expressions for the aberration integrals are used, it is not possible to recalculate with a sufficient accuracy the aberration at different planes, because the error introduced by using arbitrary aberration formulas is random due to differently manipulated aberration expressions . For different expressions of aberrations see refs. [1,2]. If we wish to determine the deviation in any plane after the aperture plane, i.e. by supposing that Ow(zo) = Aw(z a ) = 0, we get, for the coefficients in eq. (11), Sa = Sa(rb+Àra) - [Ka - ika - K a (Za)+ika(za )]ra, KLa = 2(Ka + ika)(rb + Âra ) - [Ta +Aa - Fa( Za) - Aa(Za)] ra , KR a = (Ka - ika)(rb+%ra ) - [Aa -iaa-Aa(Za) +iaa(Za),ra,
(16)
Fa = (Fa + Aa)(rb + Àra ) - 2 [Da - ida - Da Ga) + ida(Za)1 ra, Aa= (A a +iaa )(rb+Àra ) - [Da +ida-Da(Za) - ida(Za) ] ra, Da = (Da + ida)(rb+Àra) - [Ea- Ea(Za) ] ra ,
where the individual terms Sa to da are related to the terms So to do : Sa = So, Ka =Ko +ÀSo , Fa = Fo +2XKo +XZSo ,
Aa =A o + 2ÀK o + XS., Da =Do +2XAo +ÀFo +3ÀZKo +À3So ,
(17)
Ea =Eo +4ÀDo +4ÀZAo +2~êFo +4À3Ko +À4So , k a = ko, a a = ao + 2Ako, d a =do +Àao +~Zko . In the image plane (ra(z,) = 0) we thus get the usual expression M- ' Aw(Z,) e -,6(z ' )
=
Soaaao + 2(Ko + iko)aojßoao +
(KO -
iko a2ß 0
(Fo + Ao ) ao&,ßo
+ (Ao + ia o )ßôao + (Do + ido),8,,#o,
(18)
or M-' Ow(z,) e-' 9
=
Saaaaa + 2(Ka + ika ) aaßoaa + (Ka - Ika) aaNo + (Fa + Aa) aaß.A. +(Aa+iaa)Poaa+ (Da +lda)ßoßo "
(19)
both ref have the Third-order Coefficients the X[1], expression electrons chromatic the and aberration F0 by (z) == _ _eq = =+ object -'K -(Mlf)(Fo+Ao) differentiating -(Mlf T C(b,b), C(a,b), Ao) (27 e-'6(z) 2=giK(M4B"(z') (2MIf (Mlf being )~-iKB(z aberrations (K2M31f)B2(zt)' of with -K2 fZB(z) and deviation for chromatic coefficients, (remembering =fZB2(z)rarß )(Ko)(Do complex energy )(K0 the image [ Xao dz= deviation +)+ eq of from id (CBoiko) + Aw(z planes aberration differing iko) the -B"(z0)) functions T,8J -'O(z) (10) a-2(Do-ido) 2-universal gradient the dz, that -)) LencoM, of E0 VV lie 4iKB(zo))ra (Ao-iao) In (F0 paraxial e-'9(z,)=S*a2â by the -outside for Ithe of +A0) (M/2f3) e0V +slope ilM image z2K2(M4B(zt)Bi(zt) aberration defined Lenc trajectory from the +-of(1/f2) /plane (IIMf (1/2Mf D magnetic the +Multi-lens the *ß0ß0, in ZiK3(M4B3(Z,) ofunction ray +K*a,,,80&,, (25 mean we ) )+ + from ,2(M2B2(Z~) thus focusing -field, iic(B(z,) beam 1iic(B(z,) incan the get -B(zo)B'(zo)) the ref systems slope energy again +K*a2ß expressions [1] --B3(zo)) B(zo)), we --B2(Zo)), of begiven oB(z0)), have the used, o +F*a paraxial by reduce E0 SPECIALIZED ++V,12(iKMzlf2)B(zt) M/(2f A,into trajectory o the(2), eq )3SOFTWARE form and wecan ogiven for have be F
B.
.
49
2.3. The obtained
.
M(OW(z t
. t
t
+
o
L
R
o
+A*ß2â o (20)
where K,* KR F*
A*=-(Mlf)(Ao+iao)-(Do+ida)+(1/2f2)-ZK2(M2Bz(zt)-B2(Zo))-(iKMlf)B(Z,), D*
.)
2
(21) If in we
.
.
.8) .
.31)
.
2.4. For
.
for
Owc
(22)
with
: X=CCorb-(CDo-ICAo)ra, T=(CD.+iCAO)rb-
.
(23)
For Cco=C(a,a), CDo
(24)
CBo where C(a,8)
(25)
and CAo-
.
(26) I.
50
B Lencowi, M. Lenc / Multi-lens focusing systems
Expressing the chromatic aberration in terms of as and ,ß, we get Aw,(z) e-i9(z)= [ Xaaa+Taflo ] (AVIVr),
(27)
Xa - CCa(rb+Xra) - (CDa -1CAa)ra, Ta= (CDa+iCAa)(rb+Xra)-(CBa 'itcB(zo)Jra,
(28)
CCa = CCo , CAa = CAo, CDa - CDo + ~ cCo, CBa = CBo + 2XC Do + %Ko .
(29)
with
and
Similarly, we can write for the deviation due to the chromatic aberration in the image plane the usual expressions -'B(- , )= M-l Ow,(z,) e (CCoao + CDoflo)(AV/V,) ,
(30)
M-' àwc(z,) e-'9(_,)=(CCaaa+CDaßo)(àV/Vr)«
(31)
or
2 .5. Defocused image
Usually we have to express the deviation from the paraxial trajectory in a plane which does not coincide with the Gaussian image plane of the system . This can be done with the help of the aberration coefficients given above, and again we relate this deviation back to the object plane . The defocus (i.e. the difference of the position of the object plane z o from the plane z = z, conjugate to the reference observation plane ZR) is in the case of high magnification (i .e. for I z o - zc I << fo , fo being the focal length of the objective lens) given by the difference in position of the image and the observation plane _ 4=zo-z c =(z,-zR)/ MR 1+ zi-zR (zi -zR)/( MiMR), ZR - ZPi ~] -
32
as ( Z R - ZP,)/(ZPo - zJ = MR, MR being the magnification in the reference plane ZR, and zpo and z P, are the object and image side principal planes of the system . The image appears in focus if the beam aperture a b is less than the limit given by the screen resolution 4 5 ;
I MRdab I
< 4s,
(33)
i.e. for
ab < 3s/1 MR4 I
(34)
the image is sharp on a screen. 3. Program SYSTEM A highly interactive program, SYSTEM, has been written for the computation of a system consisting of up to six magnetic electron lenses . The program is written in Fortran 77 and can be run on PCs as well as on main-frame computers . Input data consist of system input file with input and output file names and
B. Lencovd, M Lenc / Multi-lens focusing systems
51
basic settings of the computation, separate files with axial magnetic fields of individual lenses, and run time data given from the keyboard, mainly in free format . The input magnetic fields are generally calculated by the finite element method program LENS [7], and consist of fields calculated in the linear approximation followed by, eventually, up to ten fields obtained as a result of nonlinear field computations. Test fields can be used if made according to the output formats of LENS. It is necessary that the fields are calculated with sufficient accuracy (i.e. contain a sufficient number of points). The lens excitation is evaluated from the integral of the absolute value of the axial field, and thus in the case of rotation-free doublets the excitation will be positive . This excitation integral is calculated from an interpolating cubic spline function, and thus this value may differ from the actual lens excitation for low-accuracy computations. Care must also be taken that the fields do not begin or end abruptly. The initial shift of the z-coordinates of each of the lenses is given in the system input file as well as the starting lens excitations . An overall field is computed by summing the focusing fields of individual lenses according to actual lens excitation and lens position . For saturated lenses, the axial field is evaluated from the input data by linear interpolation between the results of individual nonlinear computations . The initial conditions for computation (beam voltage and energy spread, positions of image and screen planes, screen resolution, etc .), settings of the lenses, positions of check planes, and settings of output parameters, are specified in the system input file. Data in this input file are preceded by a comment giving the meaning of each input, empty input is replaced by preset defaults. As a future development, this program will be extended by adding an automatic focusing and optimization routine. For this development first more experience with the present program must be gained. To facilitate the input of system data, a menu-driven program in Turbo Pascal similar to the input for our lens design programs [8] is being developed . The actual computation of the lens system then requires little data, which can be easily input from the keyboard . The data are requested by hints of the program . First, the action has to be specified by a one-character input from a displayed menu, such as change of lens excitation, type of computation, etc., then eventual numerical data in free format are input as requested by the program . It is also possible to change interactively computation parameters such as the positions of object or image planes, beam voltage, positions of individual lenses, and positions of reference planes . In the overall field of the magnetic lens system, the paraxial electron trajectories are evaluated by a variable step fourth-order Runge-Kutta method . The same method is also used in FOCUS [6], and it is based on the evaluation of the number of subdivisions of each field interval according to the field value and slope. The aberration integrals are evaluated by Simpson's rule integration with the same step length as used for the paraxial trajectory. The expressions for aberration integrals for the third-order geometrical aberrations do not contain second derivatives of the axial field; the expressions are compact and have a consistent form for all aberrations. The intermediate image position (i.e. the position of ra (z) = 0) together with the angular magnification, beam rotation and axial aberrations are listed . Graphical display of chosen trajectories, axial field and aberration integrals is also possible. An important concept of the program is that of reference planes. These may contain apertures, or serve to check the beam dimensions in a given plane or evaluate the position of an intermediate image and its associated aberrations. After the beam limiting aperture is found, according to its position and radius the beam size in all other check planes is evaluated . Check planes between the lenses allow to separate the contribution of individual lenses to the overall system aberrations . At present the number of reference planes is limited to 20. After evaluating the final Gaussian image plane, given by the last asymptotic cross section of the trajectory ra, the magnification and the aberrations in the image plane are calculated . The position and value of the beam-limiting aperture and the aberration coefficients according to eq. (17) are determined . From the defocus, given by the difference between the image and reference planes, the sharpness of the image is deduced. It does not present any problem to combine the aberration coefficients in expressions for the aberrations in other planes and to account for the influence of the limiting aperture. All trajectory computation programs generate a considerable amount of output . It is evidently reasonable to transfer some of the output into graphical form for display on the screen of a PC. For this I . SPECIALIZED SOFTWARE
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B. Lencoo6, M. Lenc / Multi-lens focusing systems
we use a program SGPlot, written at the TU Delft [9] . The paraxial trajectories, axial flux density, and eventually their derivatives, beam rotation, and the intermediate values of all aberration integrals are stored for all points with input axial field. With SGPlot any part of the output data can be displayed, scaled, and copied on printer or plotter. It is also possible to visualize some problems such as insufficient number of points in axial fields . The original purpose of the program was the design of an imaging system of a transmission electron microscope. The analysis of a subsystem composed of a selection of the lenses is also possible . Evaluation of the behaviour of an illumination system can also be made with the present program . We have to specify the position of the gun crossover as the object, and the position of the specimen as the reference plane where the intermediate image of the object is required . All necessary parameters to characterize the illuminating system can then be determined . 4. Example As an illustrative example of the use of SYSTEM we shall take the high-resolution projection beam lithography system of Lischke [10,11] . We shall pay attention only to the two-lens system demagnifying the image of a mask on a substrate with low distortion and chromatic error, which should allow the resolution down to 0.1 I-Lm in the field of up to 8 x 8 mm2 , limited by the field curvature and astigmatism . The prcj-clion lithography system is now installed at the TH Darmstadt [12]. The illumination part will be omitted from the present study, see e.g . ref. [13] . The first lens of the system is of quite a large diameter . For the proposed excitation of 6.4 A/V1/2 its focal length should be f = 400 mm. The second lens of the system is designed with one-fourth of the focal distance of the first lens . Its object focal point should coincide with the image focal point of the first lens. If the field of the second lens is of equal shape but scaled by a factor 4, and the lenses are working in telescopic arrangement with lens excitation of the second lens equal and of opposite sign, both isotropic
400
300
200
100
0
-100
-200 -400
-300
-200
-100
0
100
200
300
400
500
600
z [mm]
Fig. 1 . Arrangement of the imaging lenses of the projection electron beam lithography system, together with the two basic paraxial trajectories and axial focusing fields (arbitrary units on y-axis) .
B. Lencood, M. Lenc / Multi-lens focusing systems
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0 .0025
Axial flux density Main field, ---- Shift lens, - - Fine focus,
0 .002
i
0 .0015
m
640 .0 At 40 At x20 40 At x20
0 .001
5x i 0-
-5x10 -
0
50
100
150
200
250
300
350
400
450
z [mm]
Fig. 2. Axial field of the main lens together with the fields of the fine focus lens and the shift lens, placed between the main lens coil and the magnetic circuit. Lens excitation 640 At, additional lenses with 2 X 20 A t for the shift lens, upper coil excited in opposite direction (scaled 20 X , integral of absolute value of B(z) is 23 .7 At) and 4 X 10 At for the fine focus lens, the first and the last coil excited in opposite direction (scaled 20 X, integral of absolute value of B(z) is 5.14 A t) .
and anisotropic distortions and chromatic magnification error should be cancelled . The geometry of the lenses and a schematic path of the two basic rays is shown in fig . 1 . The geometry of the second lens was not known exactly, and it was modelled for this study by scaling the dimensions of the first lens by a factor of 4 and by introducing an appropriate shift. In order to allow small movement of the position of the focal point of the first lens and its fine focusing, additional lens coils are put between the main focusing coil and the magnetic circuit of the lens. The lens fields were calculated in a dense mesh with 87 mesh points in axial direction and 91 mesh points in radial direction with the finite element program LENS using variable-step mesh [7,8] . The magnetic material of high relative permeability of a few thousands was supposed in the computation, performed only to the right of the lens symmetry plane . Fig. 2 shows the axial fields for the main and fine focusing, and antisymmetric field for small lens displacement . To avoid eventual problems with the overlap of the fields, the boundary for the finite element computations was put at the position of the focal point - this choice had no influence on the value of the field maximum . The object point of the first lens is at a distance 383 .202 mm above the center of the upper lens, chosen as a zero of our coordinate system, i.e. z o = - 383 .202. At z F = 383 .202 mm the focal point of the lens is placed with focal distance f = 400 mm . For these imaging conditions the lens excitation is 6.323 A/V 1 /2 . The center of the lower scaled lens should be at z = 479 .0025 mm, and its focal distance is 100 mm. At z, = 574.803, a 4 X demagnified image of the substrate at zo is formed. The aberration coefficients of the lens system are 68 .7 m for the spherical aberration and 1.72 m for the axial chromatic aberration, well in agreement with ref. [10] . Field curvature 0.0431, isotropic astigmatism 0 .0133 and anisotropic astigmatism 0.0161 mm -1 also agree well with the data in ref . [10]. Chromatic magnification error and distortion are compensated within the round-off error in the computation : in z, the error due to distortion was evaluated as 0.4 nm at 5 mm from the axis. To evaluate the influence of the additional coils for lens shift and fine focusing, we have evaluated the paraxial imaging properties and distortion and chromatic magnification error at a 5 mm distance from the I. SPECIALIZED SOFTWARE
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B. Lencomi, M. Lenc / Multi-lens focusing systems
Table 1 Upper part of the table - lens shift switched on and fine focus lens switched off, lower part of the table - fine focus switched on and lens shift switched off. NI - the value of excitation of the shift lens and fine focus lens m ampère turns for V, =10000 V, Az f - shift of focal point of the upper lens, Af, - change of focal distance ff = 400 mm of the first lens, Oz, - shift of image plane position, M. - angular magnification in the image plane. 8x are isotropic, 8y are anisotropic distortions m ~tm at 5 mm distance from the axis in the image plane . The system is achromatic . The change in the value of spherical aberration can be up to 5%, field curvature and astigmatism up to 10% with respect to the values given m the text . Az f
'if,
Oz,
M.
Sx [wm]
Sy [il-]
40 20 -20 -40
7.148 3 .734 -4.032 -8 .354
-0 .651 -0.163 -0 .163 -0 .651
-0 .524 -0 .252 0.234 0.449
3.9950 3.9987 3.9987 3 .9950
4.87 2.58 -2.95 -6 .30
1 .02 0 .55 -0 .62
40 20 -20 -40
-2 .137 -1 .064 1.057 2.106
-2.358 -1 .176
-0 .135 -0 .067 0.066 0130
3.9763 3.9982 4.0117 4.0232
-1 .23 -0.61 0.59 1.16
-0 .33 -0 .16 016 0.32
NI
1.169 2.330
-1 .32
axis . Their size should not exceed 0.0025%, or 0.125 [tin, if 40000 lines should be resolved . Table 1 gives only the distortion for parallel illumination of the mask . These errors mentioned in the table can be influenced by a proper choice of the illuminating beam direction, which can be modelled in SYSTEM e.g. by a choice of aperture plane. The chromatic error is for the relative energy spread 10 -° at least 100 X smaller than the distortion, the system is thus achromatic . The results obtained confirm the data in refs . [10,11] and demonstrate the possibilities of the program such as getting accurate and reliable results, using several lenses even with overlapping fields, and choosing the position of the aperture plane. 5. Conclusions The paper presented some vital extensions to the trajectory method (inclusion of slope aberrations, set of consistent aberration formulas, which are moreover easy to program) . The theory presented in the paper has been incorporated in a program SYSTEM, useful for many purposes in the design of systems of magnetic electron lenses. New important features of this program are the concept of check planes to find beam aperture or size and intermediate aberrations, the analysis of subsystems, and the incorporation of graphics . The use of the program has been illustrated by calculating the properties of the projection beam lithography system with auxiliary lenses for fine lens adjustments. Programs LENS and FOCUS are available from Delft Particle Optics Foundation as a part of the lens and deflector design package [7]. Acknowledgements To Prof. K.D . van der Mast, TU Delft, for continuous support of the project, to Prof. Tom Mulvey of Aston University, Birmingham, UK, for useful remarks and improving the text, and to Dr . H.W .P . Koop s from Research Institute of DBP TELEKOM in Darmstadt, FRG, for suggesting the example of the projection electron beam lithography system as a useful test of the program. References [1]
P.W . Hawkes and E. Kasper, Principles of Electron Optics, vol. I (Academic Press, London, 1989). [2] P.W. Hawkes, Magnetic Lens Theory, in : Magnetic Electron Lenses, ed . P.W. Hawkes (Springer, Berlin, 1982) pp . 1-57 .
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[3] P.W. Hawkes, Methods of Computing Optical Properties and Combating Aberrations for Low-Intensity Beams, in : Applied Charged Particle Optics, ed. A . Septier (Academic, New York, 1980), Adv. Electr. Electron Phys., suppl . 13A, pp . 45-157 . [4] P.W. Hawkes, Computer-Aided Design of Electron Lens Combinations, in : Image Processing and Computer-aided Design in Electron Optics (Academic, New York, 1973). [5] M.E .C . Maclachlan, Ph.D . Dissertation, University of Cambridge (1972) . [6] B . Lencovâ and M . Lenc, Optik (Stuttgart) 82 (1989) 64 . [7] B . Lencovâ, Program documentation of lens and deflector design package, Internal report Particle Optics Group, TU Delft (1990) . [8] B . Lencovâ and G . Wisselink, these Proceedings (3rd Int. Conf . on Charged Particle Optics, Toulouse, France, 1990) Nucl. Instr . and Meth. A298 (1990) 56. [9] P. van der Hamer et al ., SGPIot manual, version 6 .40, Internal report Res. Group Superconductivity, Dept . Appl. Phys ., TU Delft (1989) . [10] B . Lischke and W. Münchmeyer, Optik (Stuttgart) 50 (1978) 315 . [111 B . Lischke et al ., Optik (Stuttgart) 54 (1979) 325 . [121 H .W .P. Koops and J. Grob, Springer Series in Optical Sciences, eds . G . Smahl and D . Rudolph, vol . 43 (Springer, Berlin, 1984) p . 119. [13] H .W .P. Koops, Microelectron . Eng . 9 (1989) 217 .
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Computation of multi-lens focusing systems Bohumila Lencovâ * and Michal Lenc
Institute of Scientific Instruments, Czechoslovak Academy of Sciences, Kr6lovopolskd 147, CS - 612 64 Brno, Czechoslovakia
A computational method based on the trajectory method is presented for the evaluation of systems of magnetic electron lenses. A consistent set of aberration coefficients is derived for third-order geometrical and first-order chromatic aberrations. The coefficients can be evaluated for an arbitrary aperture position, and also for a defocused image position due to the inclusion of the aberration of the gradient. A computer program called SYSTEM incorporating these new features has been written and the use of the program is illustrated with a suitable example.
1. Introduction In the computation of a system of electron lenses such as the imaging or illumination system of a transmission electron microscope, it is highly desirable to be able to evaluate the performance of the system as a whole. This is so because the aberrations of individual lenses combine to give the overall aberration of the system . Moreover, the beam is usually defined by the position and size of apertures, and the aberrations are influenced accordingly. The observation plane (screen or film) does not usually coincide with the Gaussian image plane of the system, and the aberrations, in particular the distortion, may differ in different reference planes. There are basically two methods in use for the computation of the aberrations of a system of magnetic electron lenses (for further details see ref. [1]) : a) The matrix method, based on a general perturbation theory, which evaluates for each lens excitation a matrix defining both the paraxial image properties (the focal length and the position of principal planes) and the third-order geometrical and first-order chromatic aberrations . For a system of lenses, these matrices are computed for each lens, and the system properties are determined by multiplying the relevant lens and transfer matrices in a defined way. This approach, pioneered in electron optics by Hawkes [2,3], was applied in a computer program by Maclachlan [4,5], but the inclusion of the effect of the aperture position and image defocus were not considered . These programs were restricted to model fields only, and have never been widely available. A disadvantage of the matrix method is that it is not very transparent, as the intermediate results are not easy to interpret. It is also necessary to mix real and asymptotic lens properties for the computation of transmission electron microscopes, and there can also be some problems in the case of overlapping focusing fields of individual magnetic lenses . On the other hand, so far only in the matrix method have the aberrations of slope been included . b) The trajectory method, based on the variation of parameters . Two independent trajectories begin at the object, and the image position is determined with the help of a trajectory which starts, with unit slope, on the optical axis. In order to obtain the off-axis aberrations, a second independent trajectory is used, starting at unit distance from the axis in the object plane . The second trajectory can be defined either to have a zero slope at the object plane or to cross the axis in the aperture plane. The aberrations in the image plane are then evaluated from integrals expressed with the help of the two independent trajectories . The trajectory method has the virtue of being straightforward, and the beam parameters in an arbitrary plane can be interpreted easily . It can also be extended to handle the aberrations for an arbitrary aperture position and in an arbitrary observation plane, not only in the Gaussian image plane, by including an * Present address: Particle Optics Group, Department of Applied Physics, TU Delft, Lorentzweg 1, 2628 CJ Delft, The Netherlands . 0168-9002/90/$03 .50 0 1990 - Elsevier Science Publishers B.V. (North-Holland)
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B Lencouli, M. Lenc / Multi-lens focusing systems
additional aberration coefficient related to the aberration of the gradient, as shown below. It is then possible to evaluate from one specific set of aberration coefficients, evaluated with the aid of the second independent trajectory starting at the object parallel to the axis (i.e. supposing the aperture plane to be at the zero point of this trajectory, also called the aberrations for a system without an aperture) the set of aberration coefficients for an arbitrary aperture plane position . The aberrations can be then also expressed at an arbitrary plane behind the magnetic lens system, which enables us to handle a defocused image, and at any plane between the lenses, to separate the effect of individual lenses. We have thus derived a consistent set of formulas for the computation of aberration integrals, which uses the same general expression for all isotropic aberrations including the aberration of the gradient, and another expression for anisotropic aberrations and for chromatic aberrations . Then we can easily evaluate the aberration coefficients for an arbitrary position of reference and aperture planes. The third-order aberrations of the gradient are usually omitted in the trajectory method, but they can also be easily expressed here, as shown below. The interrelation between the various aberration coefficients can thus easily be established . We have previously derived a suitable method for fast and accurate paraxial trajectory tracing and for an estimate of lens excitation for given focusing conditions, and implemented it together with aberration computations in a program called FOCUS [6,7] . For the design of a system of magnetic electron lenses a new program called SYSTEM has been written . By using the experience obtained with FOCUS, it was possible to check the performance of the new program . 2. Theory 2.1. Basic paraxial trajectories We define the paraxial performance of the electron optical system with the help of two independent trajectories ra (z) and rb(z), which obey the paraxial ray equation r " (Z)
+ K 2B2 (Z)r(z)
= 0,
with K = 2(e/2moV, )1/2 , e/m o being the ratio of electron charge to mass, and Vr is the relativistically corrected electron beam voltage V: Vr = V(1 + 0 .978 x 10 - 6 V) .
(2)
Electrons starting at the object in a plane containing the axis (the meridional plane) rotate with this plane by an angle O(Z) = K
f-B(z) dz. ,> Z
The two basic trajectories ra(z) and rb(z) can be conveniently defined by their initial coordinates in the object plane z = zo : ra( Z.) =0 ' rb(Z.) = 1 >
ra(Zo) -1, rb(zo) = 0.
The electron trajectory can also be expressed in the complex notation by its coordinates in the (x, y) plane as w(z) = x(z) + iy(z) = r(z) exp(iß(z)) either as W(Z) -
aowa(Z) +Nowb(Z) ,
(5)
or as W(Z) = ( a a +
Äßo)wa(Z)+ßoWb(Z) - aaWa(Z)+ßa Wc(Z)>
(6)
B . Lencoafi, M. Lenc / Multi-lens focusing systems
47
with X defined with the help of the trajectory coordinates in an aperture plane, z = za , thus X - - rb(Za)1ra( Za)' The new trajectory wJz), which is equal to zero in the aperture plane z = Z a , need not be calculated explicitly as it can easily be evaluated from the trajectories wa and wb. The aperture angle ao used here is the slope in the object plane only provided that the magnetic field there is equal to zero ; otherwise it is related to the generalized momentum in the rotating coordinate system ao =w ' (zo )-iKB(zo )ßo .
ßo is, as usual, the coordinate in the object plane, i.e. ßo = w(z o ) . In the aperture plane (rotated for convenience by an angle - B(za )) we have _ i6(Za) . ßa -a ra(Za)-w(Za)e_ 2.2. Third-order geometrical aberrations The deviation of the ray from the paraxial trajectory can be written with the help of aberration coefficients as AO(z) =Sa2âo + KLao,ß0âo + KRao0ß0 + Fa00 (10) ß ß0 + Aß02â0 + Dp0)2ß0, 2 or as (11) Ow(z) e- 'e(Z )=Sa2 o a +K Ra a2a ßo +Fa a a ßo ßo +A a ,(32â o a +Da lao o a aâa +KLaa a ßâ where S to D and Sa to Da are complex functions of the coordinate z, namely S=Sorb- (Ko - iko + )ra , Il
KL =2(Ko +iko)rb -(Fo +Ao +iKB(z o ))ra , KR = (Ko - iko)rb - (Ao - ia o - 2iKB(z o ))ra , F-- (F,, +A .)r b - 2(D,) - ldo + I K 2 B2 (zo)ra , A=(Ao +ik o )r b -(Do +ido -2 2B2 (zo))ra,
(12)
3 3(ZO) 1 2 B(zo)B i 3 ( ZO ) - -g1KB (z o ))ra . D= (Do+1d,))rb- (Eo + ZK (zo) + '2 1K B The functions So, Ko, Fo , Do , Eo and k o , a o and do are real functions, and for z = z,, where ra(z,) = 0 and rb(z,) = M gives the magnification M, they are the usual coefficients of the third-order aberrations related to the object plane. The additional function Eo in eq. (12) is related to the aberrations of the gradient, and thus it has no influence on the aberration in the image plane. This additional coefficient is normally not included in the treatment of the trajectory method, but it is vital for the possibility of calculating the aberrations in a plane other than the image plane, and for the evaluation of the aberrations of gradient. The functions So to do may be evaluated with the help of formulas :
So =11(a,a,a,a), Ko=I,(a,b,a,a), Fo=I,(a,b,a,b) + Ao=I,(a,a,b,b),
D. I, (a,b,b,b), Ea = I,(b,b,b,b), ko = j2(a,a), a o =2I2(a,b),
~~
foZ K2B2(z) dz, Z
(13)
do = l2(b,b),
I. SPECIALIZED SOFTWARE
B. Lencovâ, M. Lenc / Multi-lens focusing systems
48
where we have denoted Z h(a,ß,y,8) = z f llrarp - K2BZrarp)l rv rs - K2BZrvrs) zo +KZ[ B~ra rp +B(ra rß+rârß ),[B ' r,rs +B(r,rs+rYrs )~} dz,
,24, P)
= 2K fo [B(rarß
+ KZBZrarp) + -1B ' (r«rß + ra rß) - 4 dz ( B ' rarR~] dz .
(14) (15)
The symmetry properties of the integral Il imply that So , F,, and Eo are always positive . The form chosen for the aberration integrals is suitable for further numerical computation. If nonconsistent expressions for the aberration integrals are used, it is not possible to recalculate with a sufficient accuracy the aberration at different planes, because the error introduced by using arbitrary aberration formulas is random due to differently manipulated aberration expressions . For different expressions of aberrations see refs. [1,2]. If we wish to determine the deviation in any plane after the aperture plane, i.e. by supposing that Ow(zo) = Aw(z a ) = 0, we get, for the coefficients in eq. (11), Sa = Sa(rb+Àra) - [Ka - ika - K a (Za)+ika(za )]ra, KLa = 2(Ka + ika)(rb + Âra ) - [Ta +Aa - Fa( Za) - Aa(Za)] ra , KR a = (Ka - ika)(rb+%ra ) - [Aa -iaa-Aa(Za) +iaa(Za),ra,
(16)
Fa = (Fa + Aa)(rb + Àra ) - 2 [Da - ida - Da Ga) + ida(Za)1 ra, Aa= (A a +iaa )(rb+Àra ) - [Da +ida-Da(Za) - ida(Za) ] ra, Da = (Da + ida)(rb+Àra) - [Ea- Ea(Za) ] ra ,
where the individual terms Sa to da are related to the terms So to do : Sa = So, Ka =Ko +ÀSo , Fa = Fo +2XKo +XZSo ,
Aa =A o + 2ÀK o + XS., Da =Do +2XAo +ÀFo +3ÀZKo +À3So ,
(17)
Ea =Eo +4ÀDo +4ÀZAo +2~êFo +4À3Ko +À4So , k a = ko, a a = ao + 2Ako, d a =do +Àao +~Zko . In the image plane (ra(z,) = 0) we thus get the usual expression M- ' Aw(Z,) e -,6(z ' )
=
Soaaao + 2(Ko + iko)aojßoao +
(KO -
iko a2ß 0
(Fo + Ao ) ao&,ßo
+ (Ao + ia o )ßôao + (Do + ido),8,,#o,
(18)
or M-' Ow(z,) e-' 9
=
Saaaaa + 2(Ka + ika ) aaßoaa + (Ka - Ika) aaNo + (Fa + Aa) aaß.A. +(Aa+iaa)Poaa+ (Da +lda)ßoßo "
(19)
both ref have the Third-order Coefficients the X[1], expression electrons chromatic the and aberration F0 by (z) == _ _eq = =+ object -'K -(Mlf)(Fo+Ao) differentiating -(Mlf T C(b,b), C(a,b), Ao) (27 e-'6(z) 2=giK(M4B"(z') (2MIf (Mlf being )~-iKB(z aberrations (K2M31f)B2(zt)' of with -K2 fZB(z) and deviation for chromatic coefficients, (remembering =fZB2(z)rarß )(Ko)(Do complex energy )(K0 the image [ Xao dz= deviation +)+ eq of from id (CBoiko) + Aw(z planes aberration differing iko) the -B"(z0)) functions T,8J -'O(z) (10) a-2(Do-ido) 2-universal gradient the dz, that -)) LencoM, of E0 VV lie 4iKB(zo))ra (Ao-iao) In (F0 paraxial e-'9(z,)=S*a2â by the -outside for Ithe of +A0) (M/2f3) e0V +slope ilM image z2K2(M4B(zt)Bi(zt) aberration defined Lenc trajectory from the +-of(1/f2) /plane (IIMf (1/2Mf D magnetic the +Multi-lens the *ß0ß0, in ZiK3(M4B3(Z,) ofunction ray +K*a,,,80&,, (25 mean we ) )+ + from ,2(M2B2(Z~) thus focusing -field, iic(B(z,) beam 1iic(B(z,) incan the get -B(zo)B'(zo)) the ref systems slope energy again +K*a2ß expressions [1] --B3(zo)) B(zo)), we --B2(Zo)), of begiven oB(z0)), have the used, o +F*a paraxial by reduce E0 SPECIALIZED ++V,12(iKMzlf2)B(zt) M/(2f A,into trajectory o the(2), eq )3SOFTWARE form and wecan ogiven for have be F
B.
.
49
2.3. The obtained
.
M(OW(z t
. t
t
+
o
L
R
o
+A*ß2â o (20)
where K,* KR F*
A*=-(Mlf)(Ao+iao)-(Do+ida)+(1/2f2)-ZK2(M2Bz(zt)-B2(Zo))-(iKMlf)B(Z,), D*
.)
2
(21) If in we
.
.
.8) .
.31)
.
2.4. For
.
for
Owc
(22)
with
: X=CCorb-(CDo-ICAo)ra, T=(CD.+iCAO)rb-
.
(23)
For Cco=C(a,a), CDo
(24)
CBo where C(a,8)
(25)
and CAo-
.
(26) I.
50
B Lencowi, M. Lenc / Multi-lens focusing systems
Expressing the chromatic aberration in terms of as and ,ß, we get Aw,(z) e-i9(z)= [ Xaaa+Taflo ] (AVIVr),
(27)
Xa - CCa(rb+Xra) - (CDa -1CAa)ra, Ta= (CDa+iCAa)(rb+Xra)-(CBa 'itcB(zo)Jra,
(28)
CCa = CCo , CAa = CAo, CDa - CDo + ~ cCo, CBa = CBo + 2XC Do + %Ko .
(29)
with
and
Similarly, we can write for the deviation due to the chromatic aberration in the image plane the usual expressions -'B(- , )= M-l Ow,(z,) e (CCoao + CDoflo)(AV/V,) ,
(30)
M-' àwc(z,) e-'9(_,)=(CCaaa+CDaßo)(àV/Vr)«
(31)
or
2 .5. Defocused image
Usually we have to express the deviation from the paraxial trajectory in a plane which does not coincide with the Gaussian image plane of the system . This can be done with the help of the aberration coefficients given above, and again we relate this deviation back to the object plane . The defocus (i.e. the difference of the position of the object plane z o from the plane z = z, conjugate to the reference observation plane ZR) is in the case of high magnification (i .e. for I z o - zc I << fo , fo being the focal length of the objective lens) given by the difference in position of the image and the observation plane _ 4=zo-z c =(z,-zR)/ MR 1+ zi-zR (zi -zR)/( MiMR), ZR - ZPi ~] -
32
as ( Z R - ZP,)/(ZPo - zJ = MR, MR being the magnification in the reference plane ZR, and zpo and z P, are the object and image side principal planes of the system . The image appears in focus if the beam aperture a b is less than the limit given by the screen resolution 4 5 ;
I MRdab I
< 4s,
(33)
i.e. for
ab < 3s/1 MR4 I
(34)
the image is sharp on a screen. 3. Program SYSTEM A highly interactive program, SYSTEM, has been written for the computation of a system consisting of up to six magnetic electron lenses . The program is written in Fortran 77 and can be run on PCs as well as on main-frame computers . Input data consist of system input file with input and output file names and
B. Lencovd, M Lenc / Multi-lens focusing systems
51
basic settings of the computation, separate files with axial magnetic fields of individual lenses, and run time data given from the keyboard, mainly in free format . The input magnetic fields are generally calculated by the finite element method program LENS [7], and consist of fields calculated in the linear approximation followed by, eventually, up to ten fields obtained as a result of nonlinear field computations. Test fields can be used if made according to the output formats of LENS. It is necessary that the fields are calculated with sufficient accuracy (i.e. contain a sufficient number of points). The lens excitation is evaluated from the integral of the absolute value of the axial field, and thus in the case of rotation-free doublets the excitation will be positive . This excitation integral is calculated from an interpolating cubic spline function, and thus this value may differ from the actual lens excitation for low-accuracy computations. Care must also be taken that the fields do not begin or end abruptly. The initial shift of the z-coordinates of each of the lenses is given in the system input file as well as the starting lens excitations . An overall field is computed by summing the focusing fields of individual lenses according to actual lens excitation and lens position . For saturated lenses, the axial field is evaluated from the input data by linear interpolation between the results of individual nonlinear computations . The initial conditions for computation (beam voltage and energy spread, positions of image and screen planes, screen resolution, etc .), settings of the lenses, positions of check planes, and settings of output parameters, are specified in the system input file. Data in this input file are preceded by a comment giving the meaning of each input, empty input is replaced by preset defaults. As a future development, this program will be extended by adding an automatic focusing and optimization routine. For this development first more experience with the present program must be gained. To facilitate the input of system data, a menu-driven program in Turbo Pascal similar to the input for our lens design programs [8] is being developed . The actual computation of the lens system then requires little data, which can be easily input from the keyboard . The data are requested by hints of the program . First, the action has to be specified by a one-character input from a displayed menu, such as change of lens excitation, type of computation, etc., then eventual numerical data in free format are input as requested by the program . It is also possible to change interactively computation parameters such as the positions of object or image planes, beam voltage, positions of individual lenses, and positions of reference planes . In the overall field of the magnetic lens system, the paraxial electron trajectories are evaluated by a variable step fourth-order Runge-Kutta method . The same method is also used in FOCUS [6], and it is based on the evaluation of the number of subdivisions of each field interval according to the field value and slope. The aberration integrals are evaluated by Simpson's rule integration with the same step length as used for the paraxial trajectory. The expressions for aberration integrals for the third-order geometrical aberrations do not contain second derivatives of the axial field; the expressions are compact and have a consistent form for all aberrations. The intermediate image position (i.e. the position of ra (z) = 0) together with the angular magnification, beam rotation and axial aberrations are listed . Graphical display of chosen trajectories, axial field and aberration integrals is also possible. An important concept of the program is that of reference planes. These may contain apertures, or serve to check the beam dimensions in a given plane or evaluate the position of an intermediate image and its associated aberrations. After the beam limiting aperture is found, according to its position and radius the beam size in all other check planes is evaluated . Check planes between the lenses allow to separate the contribution of individual lenses to the overall system aberrations . At present the number of reference planes is limited to 20. After evaluating the final Gaussian image plane, given by the last asymptotic cross section of the trajectory ra, the magnification and the aberrations in the image plane are calculated . The position and value of the beam-limiting aperture and the aberration coefficients according to eq. (17) are determined . From the defocus, given by the difference between the image and reference planes, the sharpness of the image is deduced. It does not present any problem to combine the aberration coefficients in expressions for the aberrations in other planes and to account for the influence of the limiting aperture. All trajectory computation programs generate a considerable amount of output . It is evidently reasonable to transfer some of the output into graphical form for display on the screen of a PC. For this I . SPECIALIZED SOFTWARE
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B. Lencoo6, M. Lenc / Multi-lens focusing systems
we use a program SGPlot, written at the TU Delft [9] . The paraxial trajectories, axial flux density, and eventually their derivatives, beam rotation, and the intermediate values of all aberration integrals are stored for all points with input axial field. With SGPlot any part of the output data can be displayed, scaled, and copied on printer or plotter. It is also possible to visualize some problems such as insufficient number of points in axial fields . The original purpose of the program was the design of an imaging system of a transmission electron microscope. The analysis of a subsystem composed of a selection of the lenses is also possible . Evaluation of the behaviour of an illumination system can also be made with the present program . We have to specify the position of the gun crossover as the object, and the position of the specimen as the reference plane where the intermediate image of the object is required . All necessary parameters to characterize the illuminating system can then be determined . 4. Example As an illustrative example of the use of SYSTEM we shall take the high-resolution projection beam lithography system of Lischke [10,11] . We shall pay attention only to the two-lens system demagnifying the image of a mask on a substrate with low distortion and chromatic error, which should allow the resolution down to 0.1 I-Lm in the field of up to 8 x 8 mm2 , limited by the field curvature and astigmatism . The prcj-clion lithography system is now installed at the TH Darmstadt [12]. The illumination part will be omitted from the present study, see e.g . ref. [13] . The first lens of the system is of quite a large diameter . For the proposed excitation of 6.4 A/V1/2 its focal length should be f = 400 mm. The second lens of the system is designed with one-fourth of the focal distance of the first lens . Its object focal point should coincide with the image focal point of the first lens. If the field of the second lens is of equal shape but scaled by a factor 4, and the lenses are working in telescopic arrangement with lens excitation of the second lens equal and of opposite sign, both isotropic
400
300
200
100
0
-100
-200 -400
-300
-200
-100
0
100
200
300
400
500
600
z [mm]
Fig. 1 . Arrangement of the imaging lenses of the projection electron beam lithography system, together with the two basic paraxial trajectories and axial focusing fields (arbitrary units on y-axis) .
B. Lencood, M. Lenc / Multi-lens focusing systems
53
0 .0025
Axial flux density Main field, ---- Shift lens, - - Fine focus,
0 .002
i
0 .0015
m
640 .0 At 40 At x20 40 At x20
0 .001
5x i 0-
-5x10 -
0
50
100
150
200
250
300
350
400
450
z [mm]
Fig. 2. Axial field of the main lens together with the fields of the fine focus lens and the shift lens, placed between the main lens coil and the magnetic circuit. Lens excitation 640 At, additional lenses with 2 X 20 A t for the shift lens, upper coil excited in opposite direction (scaled 20 X , integral of absolute value of B(z) is 23 .7 At) and 4 X 10 At for the fine focus lens, the first and the last coil excited in opposite direction (scaled 20 X, integral of absolute value of B(z) is 5.14 A t) .
and anisotropic distortions and chromatic magnification error should be cancelled . The geometry of the lenses and a schematic path of the two basic rays is shown in fig . 1 . The geometry of the second lens was not known exactly, and it was modelled for this study by scaling the dimensions of the first lens by a factor of 4 and by introducing an appropriate shift. In order to allow small movement of the position of the focal point of the first lens and its fine focusing, additional lens coils are put between the main focusing coil and the magnetic circuit of the lens. The lens fields were calculated in a dense mesh with 87 mesh points in axial direction and 91 mesh points in radial direction with the finite element program LENS using variable-step mesh [7,8] . The magnetic material of high relative permeability of a few thousands was supposed in the computation, performed only to the right of the lens symmetry plane . Fig. 2 shows the axial fields for the main and fine focusing, and antisymmetric field for small lens displacement . To avoid eventual problems with the overlap of the fields, the boundary for the finite element computations was put at the position of the focal point - this choice had no influence on the value of the field maximum . The object point of the first lens is at a distance 383 .202 mm above the center of the upper lens, chosen as a zero of our coordinate system, i.e. z o = - 383 .202. At z F = 383 .202 mm the focal point of the lens is placed with focal distance f = 400 mm . For these imaging conditions the lens excitation is 6.323 A/V 1 /2 . The center of the lower scaled lens should be at z = 479 .0025 mm, and its focal distance is 100 mm. At z, = 574.803, a 4 X demagnified image of the substrate at zo is formed. The aberration coefficients of the lens system are 68 .7 m for the spherical aberration and 1.72 m for the axial chromatic aberration, well in agreement with ref. [10] . Field curvature 0.0431, isotropic astigmatism 0 .0133 and anisotropic astigmatism 0.0161 mm -1 also agree well with the data in ref . [10]. Chromatic magnification error and distortion are compensated within the round-off error in the computation : in z, the error due to distortion was evaluated as 0.4 nm at 5 mm from the axis. To evaluate the influence of the additional coils for lens shift and fine focusing, we have evaluated the paraxial imaging properties and distortion and chromatic magnification error at a 5 mm distance from the I. SPECIALIZED SOFTWARE
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B. Lencomi, M. Lenc / Multi-lens focusing systems
Table 1 Upper part of the table - lens shift switched on and fine focus lens switched off, lower part of the table - fine focus switched on and lens shift switched off. NI - the value of excitation of the shift lens and fine focus lens m ampère turns for V, =10000 V, Az f - shift of focal point of the upper lens, Af, - change of focal distance ff = 400 mm of the first lens, Oz, - shift of image plane position, M. - angular magnification in the image plane. 8x are isotropic, 8y are anisotropic distortions m ~tm at 5 mm distance from the axis in the image plane . The system is achromatic . The change in the value of spherical aberration can be up to 5%, field curvature and astigmatism up to 10% with respect to the values given m the text . Az f
'if,
Oz,
M.
Sx [wm]
Sy [il-]
40 20 -20 -40
7.148 3 .734 -4.032 -8 .354
-0 .651 -0.163 -0 .163 -0 .651
-0 .524 -0 .252 0.234 0.449
3.9950 3.9987 3.9987 3 .9950
4.87 2.58 -2.95 -6 .30
1 .02 0 .55 -0 .62
40 20 -20 -40
-2 .137 -1 .064 1.057 2.106
-2.358 -1 .176
-0 .135 -0 .067 0.066 0130
3.9763 3.9982 4.0117 4.0232
-1 .23 -0.61 0.59 1.16
-0 .33 -0 .16 016 0.32
NI
1.169 2.330
-1 .32
axis . Their size should not exceed 0.0025%, or 0.125 [tin, if 40000 lines should be resolved . Table 1 gives only the distortion for parallel illumination of the mask . These errors mentioned in the table can be influenced by a proper choice of the illuminating beam direction, which can be modelled in SYSTEM e.g. by a choice of aperture plane. The chromatic error is for the relative energy spread 10 -° at least 100 X smaller than the distortion, the system is thus achromatic . The results obtained confirm the data in refs . [10,11] and demonstrate the possibilities of the program such as getting accurate and reliable results, using several lenses even with overlapping fields, and choosing the position of the aperture plane. 5. Conclusions The paper presented some vital extensions to the trajectory method (inclusion of slope aberrations, set of consistent aberration formulas, which are moreover easy to program) . The theory presented in the paper has been incorporated in a program SYSTEM, useful for many purposes in the design of systems of magnetic electron lenses. New important features of this program are the concept of check planes to find beam aperture or size and intermediate aberrations, the analysis of subsystems, and the incorporation of graphics . The use of the program has been illustrated by calculating the properties of the projection beam lithography system with auxiliary lenses for fine lens adjustments. Programs LENS and FOCUS are available from Delft Particle Optics Foundation as a part of the lens and deflector design package [7]. Acknowledgements To Prof. K.D . van der Mast, TU Delft, for continuous support of the project, to Prof. Tom Mulvey of Aston University, Birmingham, UK, for useful remarks and improving the text, and to Dr . H.W .P . Koop s from Research Institute of DBP TELEKOM in Darmstadt, FRG, for suggesting the example of the projection electron beam lithography system as a useful test of the program. References [1]
P.W . Hawkes and E. Kasper, Principles of Electron Optics, vol. I (Academic Press, London, 1989). [2] P.W. Hawkes, Magnetic Lens Theory, in : Magnetic Electron Lenses, ed . P.W. Hawkes (Springer, Berlin, 1982) pp . 1-57 .
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[3] P.W. Hawkes, Methods of Computing Optical Properties and Combating Aberrations for Low-Intensity Beams, in : Applied Charged Particle Optics, ed. A . Septier (Academic, New York, 1980), Adv. Electr. Electron Phys., suppl . 13A, pp . 45-157 . [4] P.W. Hawkes, Computer-Aided Design of Electron Lens Combinations, in : Image Processing and Computer-aided Design in Electron Optics (Academic, New York, 1973). [5] M.E .C . Maclachlan, Ph.D . Dissertation, University of Cambridge (1972) . [6] B . Lencovâ and M . Lenc, Optik (Stuttgart) 82 (1989) 64 . [7] B . Lencovâ, Program documentation of lens and deflector design package, Internal report Particle Optics Group, TU Delft (1990) . [8] B . Lencovâ and G . Wisselink, these Proceedings (3rd Int. Conf . on Charged Particle Optics, Toulouse, France, 1990) Nucl. Instr . and Meth. A298 (1990) 56. [9] P. van der Hamer et al ., SGPIot manual, version 6 .40, Internal report Res. Group Superconductivity, Dept . Appl. Phys ., TU Delft (1989) . [10] B . Lischke and W. Münchmeyer, Optik (Stuttgart) 50 (1978) 315 . [111 B . Lischke et al ., Optik (Stuttgart) 54 (1979) 325 . [121 H .W .P. Koops and J. Grob, Springer Series in Optical Sciences, eds . G . Smahl and D . Rudolph, vol . 43 (Springer, Berlin, 1984) p . 119. [13] H .W .P. Koops, Microelectron . Eng . 9 (1989) 217 .
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