Electron-electron interaction induced beam displacement in a multiple electron beamsystemMing L. Yu, Steven T. Coyle, William DeVore, and Bassam Shamoun

Citation: Journal of Vacuum Science & Technology B 23, 2589 (2005); doi: 10.1116/1.2101788 View online: http://dx.doi.org/10.1116/1.2101788 View Table of Contents: http://scitation.aip.org/content/avs/journal/jvstb/23/6?ver=pdfcov Published by the AVS: Science & Technology of Materials, Interfaces, and Processing Articles you may be interested in Writing strategy and electron-beam system with an arbitrarily shaped beam J. Vac. Sci. Technol. B 22, 3543 (2004); 10.1116/1.1826057 Development of an electron-beam lithography system for high accuracy masks J. Vac. Sci. Technol. B 21, 823 (2003); 10.1116/1.1547725 Development of data conversion system for electron beam projection lithography J. Vac. Sci. Technol. B 20, 2672 (2002); 10.1116/1.1520571 Electron beam-induced carbon masking for electrodeposition on semiconductor surfaces Appl. Phys. Lett. 78, 2940 (2001); 10.1063/1.1371243 Mask electron modeling for Coulomb interaction by mask-scattered electrons in electron-beam projectionlithography J. Vac. Sci. Technol. B 16, 3227 (1998); 10.1116/1.590356

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Electron-electron interaction induced beam displacement in a multipleelectron beam system

Ming L. Yu, Steven T. Coyle,a� William DeVore, and Bassam Shamounb�

Etec Systems, Applied Materials, Inc., 26460 Corporate Avenue, Hayward, California

�Received 1 June 2005; accepted 6 September 2005; published 30 November 2005�

We have studied electron-electron �e-e� interaction induced beam displacement in multiple electronbeam systems both experimentally and using trajectory simulation. A prototype 32-beamlithography system was used to record the beam displacements on chrome mask plates and the latterwere directly measured by electron microscopy. Both experimental data and simulation results ofdemagnifying columns were consistent with a demagnification of the electron beam array withincreasing current. While the simulation matched qualitatively the experimental data well, itpredicted a smaller effect than measured. The overall demagnification was underestimated by 20%–30%. In an attempt to understand the underlying physical reason, we used a phenomenologicalmodel for the e-e induced beam displacement to fit the acquired data. The analysis suggested thatthe trajectory simulation underestimated the defocusing lens effect of the interacting electrons byabout 1.5. The parametric expression derived from this model may be used to correct for globalspace charge effects during multi-electron beam lithography. © 2005 American Vacuum

Society. �DOI: 10.1116/1.2101788�

I. INTRODUCTION

The placement of the electron beam is critical in electron-beam lithography. This article focuses on the issue of beamdisplacement by electron-electron �e-e� interaction in multi-electron beam systems. The effects of e-e interaction havebeen the subject of much investigation in recent years.1–6 It iswell known that stochastic interactions among the electronscan cause a broadening of the longitudinal energy distribu-tion of the electrons in the beam �Boersch effect�,1 and canalso cause trajectory displacements that lead to a broadeningof the beam diameter �Loeffler effect�.1 The global spacecharge effect, on the other hand, causes the focusing tochange in the column.1 Mankos et al.6 have evaluated thespace charge defocusing effect on beam displacement in amultibeam lithography column using Monte Carlo simula-tions. For a single Gaussian beam column, the defocusingcaused by the space charge can be corrected by readjustmentof the objective lens. In a multi-beam system, the spacecharge results in a demagnification of the beam array whichis current dependent. In addition, nonuniform current densi-ties result in higher-order beam deflections. In this article, wereport an experimental measurement of this beam displace-ment effect and compare the data with results from trajectorysimulations.

At Etec, we developed a prototype 50 kV raster multi-beam lithography tool suitable for mask pattern generation.7,8

This tool has a linear array of 32 electron beams in a singlecolumn. The goal of our experiment was to use lithography

a�Author to whom correspondence should be addressed; present address:Gatan, Inc., Pleasanton, CA; electronic mail: scoyle@gatan.com

b�

Present address: AKT, Applied Materials, Santa Clara, CA.

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to directly measure the e-e induced beam displacements. Thedata were then compared with the predictions from Munro’sMonte Carlo trajectory simulations.9

II. EXPERIMENT

A. Concept of the experiment

Lithography provides a direct measure of the e-e inducedbeam displacement. We selected five beams which spreadacross the 32-beam array as probe beams. These probebeams were used to write five lines on a chrome mask plate.During the scanning of these beams, a selected group ofbeams of the array was turned on. The onset of e-e interac-tion would displace the position of these probe beams. Theamount of displacement would be recorded by the lines writ-ten by these beams on the mask.

According to our simulation, the beam displacement de-pends on the currents of all the beams and the position of theprobe beam in the array. We have chosen beams 1, 9, 16, 24,and 32 as probe beams. The procedure was the following.

�1� With the linear array in the X direction, the five probebeams were turned on. Writing of the pattern on thechrome mask was by done by scanning the array in the Ydirection.

�2� At the midpoint of the Y scan, a selected set of beams ofthe array was switched on rapidly and remained on forthe rest of the Y scan. The current distribution in thearray is called a pattern in this article.

�3� After the development and etching of the mask plate,shifts in the positions of the lines written by the probebeams were measured at the point where beam switchingoccurred. The shifts were the e-e induced displacements

of the probe beams.

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We used five different array patterns. Figure 1 shows theelectron micrographs of these five patterns on the chromemask. The five long lines in the vertical direction were writ-ten by the probe beams. We shall use the convention thatbeam 1 is the edge beam on the right and beam 32 is theedge beam on the left of the array. The probe beams dividedthe rest of the array into four groups of beams: 2–8, 10–15,17–23 and 25–31. In pattern 1, all four groups of the arraywere turned on for the e-e interaction. Patterns 2 and 3 werealso quasi-symmetric patterns. In pattern 2, the two outerbeam groups �1, 4� were turned on. In pattern 3, the twoinner groups �2, 3� were turned on. Pattern 4 and 5 wereasymmetric patterns. In pattern 4, groups 2 and 4 wereturned on. In pattern 5, groups 3 and 4 were turned on.

B. Apparatus

The details of our 32-beam lithography system were re-ported earlier.7,8 A 257 nm laser beam was split into 32beams, each of which was individually modulated by anacousto-optic modulator �AOM�. This array was focusedonto a CsBr photocathode10 to generate the electron beams.The array was 65.1 �m long and the spot size was �300 nmfull width at half maximum �FWHM� at the photocathode.The array of electron beams was accelerated to 50 kV andfocused onto and scanned across a mask electron optically.Each beam could be turned on and off by its AOM with arise/fall time of about 9 ns. The nominal beam spot size wasabout 50 nm. The beam-beam separation was 250 nm andthe length of the array was 7.75 �m at the writing plane. Thebeam current was typically a few nA per beam.

C. Experimental details

Printing was done on a standard 6 in�6 in chrome plate.The resist was 300 nm of REAP 200 positive resist with anominal dose of �20 �C/cm2 for isolated features. Thechrome thickness was 70 nm. For each of the five patterns,we printed with five different doses and 21 different focussettings. The beam array was aligned in X and the scan di-rection was in Y, so all lines were Y lines. Each printedpattern covered 7.75 �m�325 �m. The periodicity of thepattern was 12.8 �m in X, and 500 �m in Y. The pixel ratewas 50 MHz and the pixel size in Y was about 40 nm. Print-ing was followed by postexposure bake and development.The resist pattern was examined by a KLA-Tencor CD scan-ning electron microscope �SEM� 8100. If the print was ac-

ceptable, it was followed by dry chrome etch.

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The current in each beam was the product of the laserintensity of the beam spot at the photocathode, the pho-toyield of the latter, and the column transmission. The majorfactor in beam current uniformity was the photoyield, whichwas not uniform across the photocathode. Beam current uni-formity could be achieved by adjusting the laser intensities atthe spots. We did not have adequate electronics for indi-vidual beam intensity control for all 32 beams. While thiswould be a problem for lithography, it was compensated forin this study by measuring the actual current of each beamprior to printing and using these values in the correspondingsimulations. We used the total laser power to adjust the totalarray current. The current per beam was �3–5 nA. To cor-rect for possible drift of beam current during the print, wealso measured the change of the total array current beforeand after the print. The average value was used for driftcorrection of the individual beam currents for our trajectorysimulations. The averaged drift correction was �3.9% andthe maximum was �5.5%.

Since we were printing lines with nonoverlapping Gauss-ian beams, the linewidth in X was controlled essentially bythe spot size and the beam current of the individual beam.The consequence of beam current nonuniformity was that thelinewidths in the prints were not uniform. Every pattern wasprinted with a dose series using 5, 6, 7, 8, 10, 12, 14, and 16passes for the dose variation. Patterns that were useful camemostly from 12 and 14 passes. These prints had less under-exposure or excessive over exposure.

To compensate for the variations in the height of the maskplate, every pattern was printed with 21 focus settings cen-tered on the nominal focus setting. The nominal objectivesetting was established by measuring the spot diameter ofbeam 16 using the beam-on-edge technique with a tungstencoated silicon grid. The nominal beam spot sizes in both Xand Y directions for this experiment were around 70 nm.

Charging was a concern for this experiment. If the beamsor their backscattered or secondary electrons hit an insulatingsurface, there can be charging either in the column or in thevicinity of the writing area. When the array beams wereturned on, the charge would build up with the rise time T0 ofthe beams and dissipate with a time constant �. In addition tothe e-e effect, the probe beams could shift in position due tocharging. To minimize the effect of charging, we need toswitch the beam very rapidly �small T0� and measure the e-einduced displacement in a time much less than �. This cor-responds to a length along the printed line determined by thepixel time and size. We controlled the beam switching

FIG. 1. Electron micrographs of the five patterns writtenwith the 32 beam array. Micrographs �a�–�e� correspondto patterns 1–5, respectively. The field is 12 �m�12 �m. The five probe beams were on all the time asthe array was scanned upward in the Y direction. Theother beams were turned on at about the middle of thefield.

through the input voltages to the AOMs. The rise and fall

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time of these control voltages was �5 ns. The AOMs havean intrinsic rise/fall time �9 ns. In practice, we found thatthe switching of the electron beams was not simultaneous.The reason was a combination of electronics delay and cablelength differences. The error was �2 pixel time or �40 ns.This translated to �80 nm on the print. This can be seen inFig. 2 as the lines do not all end coincidentally.

III. TRAJECTORY SIMULATIONS

One of the major objectives of this study is to evaluate theaccuracy of direct ray tracing software. The program Imageby MEBS Ltd.9 was used. It is a Monte Carlo simulationpackage that computes the combined effects of aberrationsand Coulomb effects. The fields in the gun extraction regionand the magnetic lenses were modeled using the second or-der finite element method �SOFEM by MEBS Ltd.�. Thecolumn was then modeled in IMAGE using the output filesfrom SOFEM and the actual column geometry. The beamemission geometry at the cathode was approximated by alinear array of 32 square beams of uniform intensity �acrosseach square� and 300 nm size separated by 2.1 �m. Theamount of current in each beam was set independently usingthe actual current measured during the print. The actualbeams in the system are approximately round Gaussianbeams of 300 nm FWHM. This approximation is justifiedsince we were primarily interested in the relative deflectionsof these beams. Off-axis behavior due to scanning the arrayduring lithographic printing was not modeled. The main scanis perpendicular to the beam array, while scanning in theother axis is typically held to less than about ±10 �m. As aresult, distortion of the array due to off-axis aberrations issmall. In addition, the beam switching �and displacementmeasurements� occurred approximately at scan center.

The position of each beam at the final plane was calcu-lated as the center of mass of all electrons belonging to thatbeam. Two simulations were performed for each arrange-ment of beams: one with only the probe beams, and one withall the beams in that pattern. The calculated shift of eachprobe beam due to e-e interactions is the position differenceof that beam between these two simulations. In this manner

FIG. 2. Electron micrographs of resist patterns featuring the five long linesprinted by the five probe beams in pattern 1. The numbers underneath arethe beam numbers. Each micrograph covers 1.5 �m�1.5 �m. The twoprobe beams 1 and 32, in the left-most and right-most pictures, respectively,show large shifts on the order of 30 nm when the array beams were turnedon as shown by the shorter lines. Probe beams 9 and 24 showed about10 nm shifts. Probe beam 16 at the center showed only a small shift.

the experimental conditions were approximately duplicated.

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IV. EXPERIMENTAL OBSERVATIONS

A. Image analysis

We observed e-e induced beam displacements in all ourprints. The electron micrographs of the resist patterns in Fig.2 show the e-e induced beam position shift of the five probebeams in pattern 1 �full array�. Each micrograph covers1.5 �m�1.5 �m. The long lines were written by the probebeams. The beam numbers count from right to left. The fullarray was switched on at about the middle of the field. Thetwo probe beams 1 and 32 had large e-e induced shifts on theorder of 30 nm. Probe beams 9 and 24 showed about 10 nmshifts. The shift of probe beam 16 at the center was afew nm.

We examined both resist patterns and chrome patternswith a KLA-Tencor 8100 CD-SEM. Owing to resist charg-ing, we found that the data from the chrome patterns weremore reliable and we used them in this article. We were notable to automate the line position measurements, therefore,we deduced the beam displacements from SEM micrographs.All the micrographs for measurements were taken at 100 kXwith an estimated beam diameter of 5 nm at 1 kV. All thepictures were 480�480 pixels at 3.125 nm per pixel. Therandom digitization error is 1 /�12 of this value11 or 0.9 nmwhich is acceptable.

We used the MATLAB© image analysis module12 for theline position determination. The middle of the line beforeand after the switching of the array beams was determinedand the e-e induced shift measured by the shift of the linecenter for each probe beam. We used the line center to mini-mize any complexity from the focus change and proximityeffect. We found that the biggest source of error came fromnoise induced line-edge roughness. We determined the e-einduced shifts by averaging the line center position over�200–300 nm on both sides of the transition. We averagedthe e-e induced beam shift values from five different printsand estimated the standard deviation.

B. Column alignment

Beam alignment in the column can change the e-e in-duced beam displacement. We have observed asymmetry inthe e-e induced shifts even with a relatively symmetric pat-tern. A misalignment of the beam array from the column axiscan create an additional shift in the e-e induced beam dis-placements that is uniform across the array. While it wouldchange the absolute values of the shifts, the relative shiftsamong the probe beam would not be affected. This, we think,is the most plausible explanation of the asymmetry we ob-served. Since we did not have a convenient gauge to evaluatethe alignment of the array, we have to assume a finite mis-alignment. Therefore, when we made comparisons to simu-lation data where perfect alignment was assumed, we uni-formly shifted the experimental displacement data by anamount that gave the least square fit to the simulation data.

Such shifts were usually on the order of a couple of nm.

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C. e-e induced focus change

We have observed changes in spot size at the onset of e-einteraction, as evidenced by changing linewidth. The objec-tive lens current could be adjusted to minimize the effect ofe-e interaction on the linewidth by setting the current mid-way between the optimum focus for 1 beam and for 32beams. It is known that e-e interaction can increase the chro-matic aberration through a current dependent increase in theenergy spread of the electrons.1 It can also increase the beamblur through the stochastic shift in lateral position and lateralvelocity of the electrons,1 and there can be a change in focusposition through the space charge effect.1 We did not have ameans to distinguish the different e-e contributions to thelinewidth. Even though we have not seen any evidence that alinewidth change would alter the e-e induced beam displace-ment, to avoid any possible complications from the focuschange, we have chosen to analyze only those data where thelinewidth change was not significant.

D. Dependence on pattern

The e-e induced beam displacements have a strong depen-dence on the array pattern. The displacement did not relate tothe presence of beams in its neighborhood in a simple fash-ion. A good example is probe beams 9 and 24 in the asym-metric pattern 2. They were symmetrically placed in the ar-ray. However, while beam 24 was nested in the middle of 16beams, beam 9 had no beam around it for 6 array spacing atboth sides. In this case, probe beam 9 had more than twicethe displacement of beam 24.

E. Comparison between experiment and simulation

We make the following general observations.

�1� The qualitative features matched well between the ex-perimental observations and the trajectory simulationsespecially for the direction of shift for the outmost probebeams 1 and 32. That they always moved towards eachother signified a demagnification of the length of thearray by the e-e interactions. The values of these relativemotions are listed in Table I. This is consistent with theinitial study of this effect by Mankos et al.6

�2� The relative displacements between probes 1 and 32 aspredicted by the trajectory simulations were consistentlylower than those observed in the experiment. The simu-

TABLE I. Comparison of measured and simulated rel

Pattern

Relative shifts �nm�

Measured Simulated

1 62.2 45.12 35.6 26.73 24.2 16.64 33.4 24.65 30.8 22.9

lated results were 20%–30% less than the measured val-

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ues as shown in Table I. The differences were more thanan order of magnitude larger than the standard deviationof the measurements and cannot be accounted for byexperimental uncertainties. Analysis to be published laterindicate that these discrepancies were not a simple scal-ing of the simulation data, but rather an underestimationof the divergent lens effect.

V. DISCUSSIONS

A. Model

While the experimental data and the trajectory simulationresults showed good qualitative agreement, it is important tounderstand the discrepancies between them. The beam dis-placements were not simple functions of the distance fromthe column axis. It is therefore difficult to learn what thephysical nature of the discrepancies is. To assist in our un-derstanding, we introduce a simple phenomenological ana-lytic expression with four adjustable parameters to analyzeboth the experimental and the simulation data. The develop-ment of the analytical expression was guided by physicalreasoning. While it is not a rigorous model of the column, ithelps us to gain an insight into the effects of e-e interaction.Here we shall briefly state the ideas behind it.

1. On-axis divergent lens

The space charge in an electron beam has the effect of adivergent lens.1 The demagnification induced shift is in-versely proportional to the total array beam current. We pa-rametrize it by writing:

Shift of beam “ i ” = ���k

Ikxi. �1�

Here Ik is the current in beam “k.” xi is the distance of beami from the column axis. � is a scaling parameter.

2. Effect of an off-axis divergent lens

If the pattern is not symmetric, the electron distribution inregions above and below the limiting aperture will also beasymmetric. We approximate the effect of these regions by adivergent lens with a shifted axis. In addition to the demag-nification discussed, there is a shift of all the beams that is

shifts between probe beams 1 and 32.

Measured-simulated �nm� Simulated/measured

17.1 0.739.0 0.757.6 0.698.6 0.747.9 0.74

ative

proportional of D / f , where f is the focal length. We approxi-

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mate the position of the lens axis by the position of thecentroid of the beams: D=�kIkxk /�kIk. Since 1/ f is propor-tional to sum of the currents of all beams,

the overall shift = �D�k

Ik = ��k

Ikxk, �2�

where � is our scaling parameter.

3. Repulsion among beams

As the beams move away from the limiting aperture, theygradually separate. The divergent lens description is not agood approximation anymore. Beams will be deflected byeach other. This deflection affects the formation of the imageinside the column. Consider the formation of a real image bya convergent lens. If we add a divergent lens on the objectside of the convergent lens, the final image will be demag-nified. If we add a divergent lens on the image side of theconvergent lens, the final image will be magnified. Theoreti-cally, we should integrate over the whole trajectory. Thissuggests the possibility of a judicious column design thatminimizes this effect.

Consider two nonoverlapping parallel beams of electronsat distance r apart. Let �1 be the charge per unit length ofbeam 1. The electrons in beam 2 will be subjected to theelectric field from beam 1. They will be deflected laterally by�1L /4�r�0V0 in distance L. Note that the deflection is pro-portional to the current of the first beam through �1, andinversely proportional to the distance r between the beams.Following the argument above, the shift of beam i by beam“j” can be approximated to a constant factor by the ratio ofthe current of beam j �Ij� to the distance between the twobeams �xij�, i.e., �Ij /xij in the first order. The argumentabove is for line charges without overlap. If they overlap,Gauss law shows that the electrons beyond xij have no effect�Faraday’s cage�. For a Gaussian beam with width , theintegral of the Gaussian beam up to xij is the correctionfactor for Ij. That factor is �1−exp�−xij

2 /22��. A larger would lower the immediate neighbor interaction. Therefore,for a cylindrical Gaussian beam j with width , the displace-ment of beam i by beam j is to within a scale factor Ij�1−exp�−xij

2 /22�� /xij. The width of the Gaussian beam is anadjustable parameter. The total displacement of beam i is

TABLE II. Sensitivity of the curve fitting of the trajec

Parameters −20%

2.52� 1.56 1.77� 1.35

therefore proportional to

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�j

Ij�1 − exp�− xij2 /22��/xij . �3�

Here i� j. We now sum the contributions from Eqs. �1�–�3�.We scale the contributions �1�–�3� and use a single scalefactor to match to the trajectory simulation data. Thereforewe have four parameters: : overall scale factor to match tothe data, �: scale factor for column axis divergent lens, �:scale factor for lens axis shift, : width of the Gaussianbeam. Displacement of the ith beam

�i = ����k

Ikxi + ���k

Ikxk+ �

j

Ij�1 − exp�− xij2 /22��/xij� . �4�

Here i� j but k includes all the beams.

4. Beam misalignment

In the above discussion, the beam array is assumed to bealigned perfectly in the column. In an actual column, thealignment may not be ideal. Let X0=the offset of the arraycenter from the axis in the X direction on the mask plane. Byintroducing this offset to Eq. �4�, we see that it simply adds aconstant shift to all beams. The relative shifts among thebeams remain unchanged. The constant shift is proportionalto ��kIk�X0. So we add another term to the shift

���k

Ik . �5�

The term � is the proportionality constant and is our fifthparameter. The complete e-e induced beam displacementanalytical expression is now

�i = ����k

Ikxi + ���k

Ikxk+ �

j

Ij�1 − exp�− xij2 /22��/xij� + ���

k

Ik �6�

with i� j.Experimentally, we can determine � by using a symmetric

array. We measure the shifts ��i’s� of two symmetric probebeams, say beams 1 and 32. Then �1+�32=2���kIk�. Sincewe know the total current �kIk, � can therefore be deter-

simulation data to the parameters in Eq. �13�.

ard deviation per beam �nm�

Best fit +20%

1.33 2.651.33 1.521.33 1.611.33 1.36

tory

Stand

mined. Unfortunately, we did not have the beam current con-

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trol to obtain a symmetric beam in our experiment. The lat-eral shift had to be compensated by a least square fitting.

5. Possible cause of disagreementbetween experiment and trajectory simulation

Our approach is to use the analytical expression to fit boththe experimental and the trajectory simulation data. By com-paring the parameters we used, we may obtain a hint on thephysical reasons for the quantitative disagreement that wesaw in Table II. We found that Eq. �6� fits the trajectorysimulation results well. The lines in Fig. 3�a� were calculatedwith Eq. �6� using the following parameters:

= 0.000 50, � = − 0.033, = 1.4, � = 0.028. �7�

Here all distances are in units of microns. The currents are innA. The symbols are the experimental data. We have notdeveloped a four-parameter automated fitting optimizationroutine. The fitting was first by eye followed by minimizingthe least square deviation for all four parameters manually.Two significant figures were used for the parameters. The

FIG. 3. �a� Comparison of the beam displacement values obtained by trajec-tory simulations �scattered points� and from the analytic expression �lines�using the set of values in Eq. �7� for the four parameters. �b� Comparison ofthe beam displacement values obtained from the experimental data �scat-tered points� and from the analytic expression �lines� using the set of valuesin Eq. �8� for the four parameters.

averaged root-mean-square �rms� deviation per point was

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1.33 nm. The largest deviation from the data was �3.4 nm.This fitting result gave us confidence that Eq. �6� is a goodguidance to our understanding of the space charge effects ofthe e-e interaction inside the column.

It is of interest to find out the sensitivity of the curvefitting to the values of the parameters. Table II compares therms deviation per point of the curve fitting if the parameterswere given ±20% variation. Parameter is the total scalingfactor and has the largest sensitivity; , which is related tobeam-beam repulsion, has the second largest sensitivity. It isfollowed by �, which is related to the on-axis divergent lenseffect of the space charge; �, which is related to the offset ofthe space charge divergent lens, is the least sensitive of all.

We made a similar curve fitting of Eq. �6� to the experi-mental data. As we discussed above, we need to make cor-rections to the possible misalignment of the beam array withrespect to the column. In the following discussion, the ex-perimental data were always shifted by least square fitting toeliminate the � term. There is one complication in this pro-cedure. For a given pattern, the � term and the � term werenot distinguishable. So we did not have a way to indepen-dently determine �. Therefore, we shall use the value of �determined from the trajectory simulation data above. It is oflittle consequence since � will absorb any error that � mighthave. The interesting observation is that we found a good fitfor all our experimental data with Eq. �6� as shown in Fig.3�b� but with a new set of parameters. They are as follows:

= 0.000 57, � = − 0.051, = 1.4, � = 0.028. �8�

The rms deviation per point was 1.2 nm. The largest discrep-ancy was about 2.4 nm.

A comparison of the two sets of parameters shows clearlythat the discrepancies between the trajectory simulations pre-dictions and the experimental observations were beyond ascaling factor. The scaling factor only increased by �14%from 0.000 50 for the simulation to 0.000 57 for the experi-mental data. The beam-beam repulsion parameter was un-changed. The major discrepancy was the parameter �, whichis related to the on-axis divergent lens effect of the spacecharge. � for the trajectory simulation decreased from−0.033 for the trajectory simulations to −0.051 for the ex-perimental data. It was �55% change. While this analysis isrudimentary, it is suggestive that the trajectory simulationshave underestimated the contributions from the space chargedivergent lens effect. For this particular column configura-tion, it was off by about a factor of 1.55.

B. Corrections

This parametric approach enables real-time correction ofglobal space charge effects during e-beam lithography. Theparameters can be determined by a combination of simula-tion and experiment, similar to this work. The gray levels ofindividual pixels may then be adjusted to place features inthe correct positions by applying the parametric expressionto each flash of the array. This model may also be extendedto two dimensions to correct for effects in two-dimensional

arrays.

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VI. CONCLUSION

We have experimentally observed e-e induced beam dis-placements using multiple e-beam lithography. By varyingthe pattern, we verified that the experimental observationsmatched very well with the predictions from the MunroMonte Carlo trajectory simulations qualitatively includingthe demagnification of the overall size of the array. Detailedanalysis showed that the trajectory simulations consistentlyunderestimated the overall demagnification by 20%–30%.

We made an attempt to use a physics based parametricexpression to determine the possible cause of the discrep-ancy. The result is indicative that the trajectory simulationsunderestimated the divergence lens effect from the spacecharge of the electron beams in this column, and the overallsize of the effect. This expression can be used for real-timecorrections of global space charge effects in a multi-electronbeam lithography system.

ACKNOWLEDGMENTS

The authors would like to thank Juan R. Maldonado,Homer Lem, Jeffrey Tran, Doug Willis, Gene Mirro, AlSagle, and Mark Gesley. This project was funded by

DARPA/SPAWAR under Contract No. N66001-99-C-8624,

JVST B - Microelectronics and Nanometer Structures

bution subject to AVS license or copyright; see http://scitation.aip.org/terms

“Multiple Electron Beams Component Prototyping for HighResolution Pattern Generators.”

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