For optical exposure equipment such as a stepper inthe fabrication of semiconductor integrated circuits,aberration in the imaging system causes a degrada-tion of the printed image quality as well as a signif-icant reduction of focus latitude of the lithographyprocess.1–3 To predict this and minimize its adverseeffects on the image, a reliable measurement tech-nique for characterizing aberration in the exposureequipment is necessary. Lens manufacturers typi-cally use high-precision interferometric tools for mea-suring the wave-front aberrations during theassembly of the projection imaging system. How-ever, interferometric techniques cannot be applied tothe fully assembled exposure system, especially afterall the lenses are mounted into the system. There-fore one can only analyze aberrated images in thephotoresist on the wafer to find out the types of ab-errations present in the imaging system.
Over the past few years, a variety of methods formeasuring aberrations in the lithographic exposuretool were proposed. In general, they belong to one oftwo categories. Early reports4–6 mainly used binaryamplitude gratings of multiple periods and orienta-tions in the chrome mask. The multiple grating pe-
The authors are with the School of Optics, CREOL, University ofCentral Florida, P.O. Box 162700, 4000 Central Florida Boulevard,Orlando, Florida 32816-2700.
Received 1 July 2002; revised manuscript received 8 November2002.
riods sample various radii in the exit pupil of theprojection optical system. Recently, techniquesbased on the use of the pure phase grating mask werereported.7,8 Depending on the type of aberrationconsidered, a shift of the best focus or lateral shift ofthe image is measured.6,9 The even aberrations,such as spherical aberration and astigmatism cause ashift of best focus depending on the shape and orien-tation of the objects in the mask. The odd aberra-tions such as coma cause placement errors and avariation of linewidth at two ends of the dense lines.1Some of these reported methods were shown to bequite accurate and in good agreement with interfero-metric measurement data. However, these methodsusually involve a complex data processing proceduresuch as curve fitting for Zernike polynomials, and�ora spectral analysis of the measured plots of focus shift�or lateral shift� versus a certain parameter. Also,higher-order aberrations beyond the fifth order arefundamentally very difficult to measure accuratelywith these proposed methods.
Therefore we were naturally led to seek the possi-bility of exploiting both amplitude and phase maskson opposing sides of the photomask for easier, moreefficient aberration characterization. Instead of us-ing multiple periods of amplitude gratings for pupilsampling, we can actually make use of multiple illu-mination settings from a phase-grating mask toachieve the pupil sampling. We discovered that theresponse of the aberrated image to each illuminationshape is slightly different. Therefore this propertycan be utilized to calculate a specific type of aberra-tion provided that those different imaging results
from multiple illuminations are measured with goodaccuracy.
2. Theoretical Background
In this section, we consider the theoretical basis ofaerial image simulation for photolithography. Basi-cally, the lithographic stepper is an imaging systemwith a partially coherent quasi-monochromatic illu-mination source.2 The optical diagram of a stepperis shown in Fig. 1. There are two important thingsto note for this diagram. First, the Fourier spec-trum of the object in the photomask is formed in thepupil plane side of the projection optics. Second, thephotomask plane receives nearly collimated illumi-nation from the condenser and an image of the sourceis formed in the pupil plane of the projection system.3The wave-front aberration is a departure of the realwave front from the ideal spherical wave front at theexit pupil of the imaging system, leading to a shiftand deformation of the image. There are two basicrepresentations of wave-front aberration, Seidel andZernike polynomials. For lens design purposes, theSeidel representation is preferred but for metrologypurpose, the Zernike orthogonal circle polynomialrepresentation is more convenient. If there is awave-front aberration W in the imaging system, thetransfer function H is modified according to
H� f, g� � H � f, g�exp�i2�
�W��, ��� , (1)
where ��, �� is a polar coordinate of the normalizedpupil coordinate �f, g�. This modified transfer func-tion was used in our aerial image calculation algo-rithm. To calculate aberrated aerial images with
given Zernike coefficients, we made a simple aerialimage simulation program. It is well known that asimple Fourier optic scalar diffraction approachworks well for the imaging systems with a small nu-merical aperture less than NA � 0.6 as explained inRef. 3. This method basically calculates a coherentimage of the object amplitude under coherent illumi-nation from a point in the source by using the trans-fer function introduced in Eq. �1� and incoherentlysums the image intensities from all source points.Compared with the more rigorous Hopkins modelfor a partially coherent imaging system,2 thismethod has the advantage of faster speed and lessmemory usage. The basic relations of relevantquantities for this aerial image calculation methodare summarized in Table 1.
According to the power of � in the wave-front ab-erration polynomial, the aberration is classifiedlargely as even and odd aberrations. Even-type ab-errations include spherical aberration, astigmatism,and field curvature, and odd-type aberrations consistof coma and distortion. The effects of these twotypes of aberration on the imagery are well explainedin Refs. 5, 6, and 9. The authors of those papers tookadvantage of the three-beam interference imagingcondition by choosing proper grating periods and us-ing a small partial coherence factor less than � � 0.2as shown in Fig. 2. Under those conditions, the ef-fects of even and odd aberrations are completely sep-arated as the focus shift and the lateral shift of theimage. In this paper, we will focus on our newmethod for characterizing even-type aberrations be-cause we do not have an overlay inspection tool,which is required to perform a measurement of therelative lateral image shift.
Table 1. Aerial Image Calculation Algorithm
Quantities considered Expressions Etc.
Image amplitude for an on-axis point source gI �x, y� � F1H�f, g� F�t�x, y�� t�x, y� is an object transparency functionObject amplitude for an off-axis point source g0 �x, y� � t�x, y�exp�i2� �fcx�gcy�� �fc, gc� is the spatial frequency from the source
Intensity of the image I�x, y� � �fc gc � gI �x, y��2
—
Spatial frequency range of the illumination f c2 � gc
2 � �2 NA2 � �2 � � NAc�NA is the partial coherence factor
Fig. 1. Optical diagram of the photolithographic stepper.
Depending on the region of the pupil sampled, dif-ferent best-focus values will be measured for evenaberrations. Sampling of the pupil can be modifiedby imaging different test objects in the chrome maskand also by changing the shape of the illuminationincident on the test object. Use of the latter variableis desirable because feature size and shape are moreor less constant for a certain device manufacturingprocess, and the lithographer wants to know aberra-tion effects associated with the objects being used.Our method is based on this observation: that mul-tiple illumination shapes can provide multiple pupilsamplings, which gives a different focus shift for thesame even aberration. Thus if we measure thesedifferent focus shifts and calculate the change of focusshift induced by a very small change of approximately0.001 � in each Zernike coefficient, it would be pos-sible to obtain the actual Zernike coefficients of theeven aberration present in the exposure tool. Toprovide multiple illumination shapes, one has to havea way of creating desired multiple illumination pat-terns incident on the conventional chrome mask.Steppers �or scanners� made for semiconductor man-ufacturing in recent years have such abilities as con-trolling the illumination shape from the condenser toenhance the resolution limit. But, for most old step-pers, the illumination shape is essentially fixed.Furthermore, even with a variable illuminator, mul-tiple exposures should be performed, one for eachillumination shape. However, if we make a properdiffractive phase profile on the backside of the con-ventional photomask, we can create different illumi-nation patterns depending on the two-dimensionalshape of the phase profile. Furthermore, we can ar-range the location of each phase profile appropriatelyin a single photomask, and this allows us to carry outonly a single exposure instead of doing several expo-sures for multiple illumination shapes, reducing thenumber of exposures required for the aberration mea-surement.
3. Our Approach to Aberration Measurement
We decided to fabricate both square binary and cir-cular binary phase gratings on the backside of thephotomask to produce the quadrupole and annularillumination shapes to provide three illuminationpatterns, which include the conventional illumina-tion. Figure 3 shows these diffractive profiles andcorresponding illumination spatial spectrums in thepupil plane. It is well known that these modifiedoff-axis illuminations improve the resolution limit ofsubwavelength features, and the spatial frequency fcof the off-axis illumination should be half of the objectfrequency f in the chrome mask,3
fc �f2
�1
2p. (2)
In the above equation, p is the period of the denselines in the chrome mask. For the pupil sampling,we designed conventional chrome gratings havingfour orientations and a proper period for three-beamimaging as shown in Fig. 4�a�. On the opposite sideof this chrome mask, we fabricated and placed aphase grating as shown in Fig. 4�b�.
Measurement of the shift of the best focus ��BF�and the difference of focus HVcos� between two per-pendicular images of the grating lines are requiredfor characterizing spherical aberration and astigma-tism, respectively. ��BF� is given as the averagefocus of four orientations of gratings, and HVcos� isgiven as the focal difference between 0, 90 deg ori-ented gratings. Using an aerial image simulationprogram, we can calculate the BF at each illumina-tion setting, corresponding to the Zernike coefficientZj of 0.001� of spherical aberration. This is essen-tially the partial derivative of ��BF� or HVcos�, withrespect to Z9. We define this quantity as the sensi-tivity S, and each value is calculated as
Fig. 2. Three-beam interference imaging through the pupil �H.Nomura, T. Sato6�.
Fig. 3. Binary phase diffractive profile and generated illumina-tion patterns at the pupil: �a� circular plane, �b� square phase.
This calculation is repeated for all illumination set-tings �i � 1 conventional, 2 quadrupole, 3 annular�and all relevant Zernike coefficients Zj, where j � 4,9, 16 for spherical aberration, and j � 4, 5, 12 forastigmatism. As a result we get a matrix of sensi-tivities, S. Then, the relevant Zernike coefficientscorresponding to spherical aberration can be ex-tracted from the following linear equations.
��BF�i ���BF�i
�Z4Z4 �
��BF�i
�Z9Z9 �
��BF�i
�Z16Z16 ,
(4)
where ��BF� is the measured shift of averaged bestfocus of four orientations of gratings, and i ��1, 2, 3�
represents one of three illumination settings. Henceif we measure ��BF� for three different illuminationconditions, then we can solve for the unknown Zernikecoefficients Z4, Z9, Z16 from the above set of linearequations. The same procedure can be applied to theastigmatism as well. To determine the best focus ofeach grating region, we take advantage of the three-beam interference imaging condition. For a given setof wavelength, NA and �, the amplitude grating periodcan be chosen so that only three propagating orders arewithin the pupil. In the case of the 50% duty cycleamplitude grating, it is given as6
�
�1 � ��NA� p �
3�
�1 � ��NA. (5)
Even though real illumination is always partiallycoherent, we ignore that for now and consider onlythe interference of three coherent waves on the im-aging plane. Let’s denote each of three beams as1, 0, �1 and phase of 1, �1 beams relative tozero-order beam as �1, �1. Then, the intensity ofimage of a dense line grating along the y axis is givenas
Fig. 4. �a� Binary chrome mask, �b� 1X plate mask for patterning diffractive profile viewed from the top.
In Eq. �6�, An represents amplitude of the nth dif-fraction order and ��z is introduced inside the lastcosine to represent the effect of defocus. The pro-portionality coefficient � of defocus is given asNA2�2�2� from the paraxial imaging theory, where �is the normalized pupil radius coordinate. If we con-sider a fixed position of x in the image, then Eq. �6�becomes a cosine function of �z and peaks at ��1 ��1��2�, which is proportional to the even aberration.In photolithography, the imaging plane is not anideal plane with zero thickness, but a layer of pho-toresist with finite thickness. The best defocus po-sition is often located toward the lenses from the topof the photoresist surface. The bottom edge of thedeveloped photoresist profile is given by a contour I�x,�z� � const. If the critical dimension �CD� is mea-sured and plotted with defocus �z for some fixedvalue of the exposure dose, then one would obtain acurve whose slope vanishes for the defocus equal to��1 � �1��2�. Depending on the threshold inten-sity level, which affects the fixed bottom edge positionx, the coefficient of this cosine like curve may bepositive or negative. Hence if we perform multipleexposures with a varying dose for each exposure re-gion, we may obtain a series of CD versus defocuscurves as in Fig. 5 that has generated from theSolid-C commercial lithography simulator. Thiscurve is called the Bossung curve.3 In reality, theturning points of these curves change slightly withthe dose, which is the result of interaction betweendefocus and finite thickness of the resist. To mini-mize this variation, one should use a thin resist.Therefore from these curves of CD versus defocus fora thin photoresist at multiple doses, one can obtainthe best defocus position for the image of the gratingconsidered. This procedure is repeated for everygrating orientation and illumination type.
4. Experiment
To demonstrate our new approach by experiment, wefirst designed two photomasks. One is the conven-tional binary chrome mask for use in the stepper, as
shown in Fig. 4�a�. The other is a 1X plate mask forpatterning the diffractive profile in the photoresist onthe backside of the chrome mask. The stepper usedin this study is a GCA wafer stepper for the G-line�436 nm� of exposure wavelength. Its magnification,numerical aperture, and partial coherence are 1�5,0.38, and 0.6, respectively. To prevent any confu-sion about the dimension, we will use the scale on thewafer for all feature sizes. Therefore the scale onthe mask is multiplied by 5. The period of the linegrating in the chrome layer was chosen to be 1.6 �m,which corresponds to the normalized pupil radius of0.7171 for the �1st order diffracted beams. The linewidth is half of the period. Four orientations of 0,90, 45, �45 deg of gratings were made in a row toaccount for the effect of 0, 90, and �45 astigmatism.
For the 1X mask for fabrication of the diffractivemask on backside of the chrome mask, its period waschosen to be twice that of the chrome grating periodaccording to Eq. �2�. To fabricate the diffractiveprofile, we spin coated Shipley 1805 photoresist onthe back of a chrome photomask with a thickness of0.315 �m corresponding to � phase shift. With aphase depth of �, unwanted zero order from thephase grating is minimized and �1 orders becomedominant illumination beams. We estimated thediffraction efficiency of the phase gratings for the�1 orders to be approximately, 40%. Also, it wasverified that the efficiency of �1 orders does notchange significantly within the thickness error of�20 nm. If the resist thickness error exceeds �20nm, it causes the unwanted zero order to rise be-yond 5% and increases the background intensitylevel in the image. That would reduce the accu-racy of the measurement, because the higher back-ground intensity would decrease the slope of theBossung curves and the turning points would bemore obscure to observe. We checked the thick-ness with our mechanical profilometer, and it wasverified to be within �20 nm around the exactvalue.
Shipley 1805 resist is used for G-line exposure, and
5. A simulated plot of line-width �CD� vs. defocus curves for many exposure doses �Bossung curves�.
is not ideal for making a phase profile. However, wefound that its transmittance at the G-line is approxi-mately 82% after postexposure baking at 125 °C for 30min in an oven. Therefore we used Shipley 1805 witha little higher exposure time to compensate for its ab-sorption loss. After coating the photoresist, wemounted both the 1X plate mask and the chrome pho-tomask in our Quintel UV-masks aligner with an I-linesource and performed the exposure. This results inthe desired surface profile for the phase grating in the1805 photoresist, as shown in Fig. 6 of the microscopeimage. The real period of these phase gratings is 16�m, which is twice that of the target grating on themask. The position of each diffractive profile wassuch that each is exactly behind one of a 1 by 3 arrayof chrome gratings grouped.
The next step was to use this fabricated hybriddiffractive photomask in our GCA stepper to do thefocus micro-step exposure. We varied the focalplane of each exposure by 0.2 �m from the previousexposure, and shifted the wafer laterally by 52 �mbetween each exposure to avoid image overlapping.The total range of defocus variation is �1 �m cen-tered on the best focus of the stepper. This isslightly larger than the Rayleigh defocus range of���2NA2� � 1.51�m. Also the exposure time wasvaried from 0.55 to 1.0 s on 3 by 3 arrayed positionson the wafer. The printed image of this focus micro-step exposure is shown in Fig. 7. The defocus isincreasing along the column direction in step of 0.2�m, and the order of variation of CD is very smallabout 0.1�0.2 �m, as predicted in our aerial imagesimulation. We used a Cambridge 360 scanningelectron microscope �SEM� to observe the variation ofCD in the printed image from focus micro-step expo-
sure. Figure 8 shows the obtained Bossung curvesfor three illumination settings from this SEM mea-surement. By taking the average and difference ofturning points for four orientations of the gratings inevery case of illumination, we obtained the shift ofbest focus ��BF� and the difference of focus HVcos�
between two perpendicular orientations of gratingsas summarized in Table 2. The error in the CDmeasurement is estimated to be �0.015 �m. Thisdata from the SEM measurement will be used toanalyze the associated even aberration as explainedin the next section.
5. Aberration Analysis
The sensitivity of best focus, as defined in Eq. �3�, iscalculated by giving a very small variation to Z9,while keeping all other Zernike coefficients zero. Itcan be obtained for all relevant Zernike coefficientsfor spherical aberration, i.e., Z4, Z9, and Z16. Thesensitivity for astigmatism is calculated by observingthe change of best focus with respect to Z4, Z5, andZ12 for two perpendicular grating objects. Then, wetake the difference of two sensitivity matrices for twoperpendicular grating objects. The resulting sensi-tivity matrices are shown in Table 3. In the vectorform, the linear relation �4� is expressed as
X � S� � Z . (7)
Here, X is a 3 by 1 vector of measured quantities,��BF� is for spherical aberration or HVcos� is for 0–90astigmatism, and S is the sensitivity matrix for eachcase. The elements of vector Z are three Zernikecoefficients for the type of aberration considered�spherical or astigmatism�. Because we measured
Fig. 6. Microscope images of the fabricated diffractive masks: �a� circular phase grating, �b� square phase grating.
every element of X, and calculated every element ofthe sensitivity matrix S, it is straightforward to solvethis equation for the unknown vector Z. Using themeasured X quantities summarized in Table 2, wesolved the above matrix Eq. �7� for Z, which is shownbelow for both spherical aberration and astigmatism.For spherical aberration,
X � ���BF�1
��BF�2
��BF�3
� � � � 0.25� 0.30� 0.10
�� � � 5.0 10.5 � 9.0
� 6.0 11.3 � 7.5� 5.0 10.5 � 7.0
� � � Z4Z9
Z16�
3 Z9 � 0.1904�, Z16 � 0.0752�
(third and fifth order). (8a)
For astigmatism,
X � �HVcos�1
HVcos�2
HVcos�3
� � �0.3620.3420.20
�� �12.08 � 8.30 5.7
12.08 � 8.56 5.912.08 � 7.64 3.1
� � � Z4Z5
Z12�
3 Z5 � 0.1706�, Z12 � 0.1068�
(third and fifth order). (8b)
Because the condition numbers for matrices in lin-ear equations �8� are quite high, values of 71 and 221respectively, we used the QR factorization method10
to obtain a stable solution for the system of linearequations of ill condition. The results for sphericalaberration and astigmatism are summarized in Table4. The errors in the Zernike coefficients due torandom variation of X elements within the mea-surement error of �0.015 �m are verified to varywithin �0.014 �, which is not very small but ac-ceptable. The values for third-order aberrationsare in good agreement with the estimated value of0.2 �, which is typical for the steppers made in themid to late 1980s. But, the value for fifth orderastigmatism is too high, compared with the esti-mated value. This too high fifth-order astigma-tism measurement is believed to be due to someerror in the measurement and the rather high par-tial coherence factor of 0.6. A smaller partial co-herence factor less than 0.3 allows almost coherentinteraction between the beams sampling differentpositions of the pupil. Hence it gives more notice-able variation of CD with defocus, ease of the mea-surement, and higher accuracy. However, we didnot attempt to reduce the source aperture.
To verify the accuracy of our measurement, weinput the obtained values of the Zernike coefficientsinto our aerial image simulation program and gener-ated the image of an equal line-space grating �Fig. 9�.We found that at the line width of approximately 0.7�m, the contrast of the image shrinks significantly,which means that 0.7 �m is the resolution limit.
Fig. 7. Microscope image of the printed focus micro-step exposure.
This agrees fairly well with the actual resolutionlimit of the stepper as specified by the manufacturer.Therefore it is a good indication that our measure-ment was somewhat accurate despite using the de-fault high partial coherence factor.
6. Conclusion
We introduced what is to our knowledge a new ap-proach to the characterization of aberration in photo-lithographic steppers. Combining binary amplitudeand a diffractive phase mask on the opposite side of amask plate allows for manipulation of the pupil sam-
Table 2. Focus Data Obtained From the Bossung Curves for ThreeIlluminations
Fig. 8. Bossung curves for three illumination cases obtained from the SEM measurement: �a� from conventional illumination, �b�quadrupole illumination, �c� annular illumination.
pling by using many different shapes of diffractive pat-terns. Measuring the focus shift under the spatiallyvarying diffracted illumination, we were able to ex-tract the associated even aberrations by solving a sys-tem of linear equations. We demonstrated this newmethod with a conventional G-Line GCA stepper andverified the accuracy of measured values of even aber-rations. The method predicted values too high for thefifth-order astigmatism, due to the measurement errorand the high sensitivity on the error caused by usinglarge partial coherence. If combined with a low par-tially coherent source, the accuracy and stability of themeasurement will be improved.
This approach has a good potential to measurehigher-order aberrations with high precision if ad-ditional numbers of diffractive illumination pat-terns are used. It is also possible to measure oddaberrations, such as coma and distortion, with thisapproach when used with properly designed targetobjects and measured with an overlay inspectiontool.
References1. T. Brunner, “Impact of lens aberrations on optical lithogra-
phy,” IBM J. Res. Dev. 41, 57–67 �1997�.
2. M. Born and E. Wolf, Principles of Optics, 6th ed. �Pergamon,Oxford, 1980�, pp. 459–490.
3. A. K. K. Wong, Resolution Enhancement Techniques in OpticalLithography, TT47 �SPIE Press, Bellingham, Wash. 2001�Chap. 2.
4. J. P. Kirk, “Measurement of astigmatism in microlithographylenses”, in Optical Lithography 11, L. V. den Hove, ed., Proc.SPIE 3334, 848–854 �1998�
5. M. S. Yeung, “Measurement of wave-front aberrations inhigh-resolution optical lithographic systems from printedphotoresist patterns,” IEEE Trans. Semicond. Manuf. 13,24–32 �2000�.
6. H. Nomura and T. Sato, “Techniques for measuring aberra-tions in lenses used in photolithography with printed pat-terns,” Appl. Opt. 38, 2800–2807 �1999�.
7. J. P. Kirk, G. Kunkel, A. K. Wong, “Aberration measurementusing in-situ two beam interferometry,” in Optical Microlithog-raphy 14, �Proc. SPIE, 4346, Bellingham, Wash., 2001� pp.8–14.
8. H. Nomura, “New phase-shift gratings for measuring aberra-tions,” in Optical Microlithography 14, �Proc. SPIE, 4346, Bel-lingham, Wash, 2001� pp. 25–35.
9. H. Nomura, K. Tawarayama, and T. Kohno, “Aberration mea-surement from specific photolithographic images: a differentapproach,” Appl. Opt. 39, 1136–1147 �2000�.
10. G. H. Golub and C. H. van Loan, Matrix Computations, 3rd ed.�Johns Hopkins U. Press, Baltimore, Md. 1996�.
Fig. 9. �a� Reconstructed wave front at the pupil, �b� simulated image of a line grating of 0.8� of the line width.
