linnik microscope imaging of integrated circuit structures · linnik microscope imaging of...

Linnik microscope imagingof integrated circuit structures

D. M. Gale, M. I. Pether, and J. C. Dainty

Experimental one-dimensional intensity and phase images of thick 1.200 nm2 oxide lines on silicon arepresented together with profiles predicted from the waveguide model. Experimental results wereobtained with a purpose-built Linnik interference microscope that makes use of phase-shiftinginterferometry for interferogram analysis. Profiles have been obtained for both TE and TM polariza-tions for a wide range of focal positions and in both bright-field 3type 11a24 scanning and confocal modes ofmicroscope operation. The results show extremely good agreement despite several simplifyingassumptions incorporated into the theoretical model to reduce computing times.Key words: Interference microscopy, waveguide imaging theory, integrated circuit metrology,

profilometry. r 1996 Optical Society of America

1. Introduction

The optical measurement of submicrometer featureshas, in the past, been extremely difficult because ofthe lack of a suitable imaging theory for thickstructures 1$200 nm2 and the absence of instrumentscapable of gathering the appropriate data. In par-ticular this has presented problems for measure-ment and inspection in the semiconductor industry,in which one commonly encounters features whoseheight and width are in the 0.1–1-µm range.Conventional scalar imaging theory has been used

successfully to provide measurements of linewidthson photomasks and thin wafers1 1#200 nm2. Animportant part of this process is the use of a primarylinewidth-measurement system to allow measure-ments to be made on real samples under well-characterized conditions consistent with those im-plied by the theory. Such systems have beendeveloped at both the National Institute of Scienceand Technology in the U.S.A.1,2 and at the NationalPhysical Laboratory in the U.K.3 With sufficientagreement between theory and experimental re-

When this research was performed the authors were with theOptics Section, Department of Physics, Imperial College of Sci-ence, Technology and Medicine, London SW7 2BZ, U.K. D. M.Gale is now with the Instituto Nacional de Astrofısica, Optica yElectronica, Apartado Postal 51 y 216, Puebla 72000, Puebla,Mexico.Received 4 May 1995; revised manuscript received 29 August

1995.0003-6935@96@010131-18$06.00@0r 1996 Optical Society of America

sults, these primary measurement systems can beused to calibrate linewidth standards, which in turncalibrate commercial measuring instrumentsthroughout the semiconductor industry.For thicker objects it becomes increasingly diffi-

cult to match experimental image profiles with thoseproduced by scalar theory; for objects $200 nm, anew theoretical approach is required. To this end,the rigorous waveguide model has been discussed inthe literature and has shown promising initial re-sults. To investigate further the validity of thewaveguide model, a primary measurement systemhas been constructed at Imperial College. The sys-tem provides phase as well as intensity images, aswe are dealing with three-dimensional 13-D2 objectsfor which the phase image may aid interpretation.In view of the considerable present interest in confo-cal microscopy it was decided to incorporate bothconfocal and nonconfocal scanned imaging modes.Principal design considerations needed to secure anappropriate primary measuring system were

c Operation in reflection for study of opaqueobjects.

c Polarization state variable between transverseelectric 1TE2 and transverse magnetic 1TM2.

c Operation at high-imaging N.A. 1numerical ap-erture2 for optimum resolution.

c Monochromatic illumination for accurate phaseinformation and high contrast.

c Kohler system for optimum bright-field illumi-nation.

c Variable illuminating N.A. for control of coher-ence.

1 January 1996 @ Vol. 35, No. 1 @ APPLIED OPTICS 131

c Laser source for high-intensity monochromaticimages.

c An effective scanning slit at the sample, whichis smaller than the impulse response width of thesystem.

c Near-diffraction-limited optics.c Axial imaging to eliminate off-axis aberrations.c Automated 1computerized2 data collection and

processing.

We describe in detail the primary measurementsystem and the phase and intensity images obtainedfor a thick silicon oxide on silicon wafer sample.These results are compared with theoretical predic-tions of the waveguide model.

2. Waveguide Imaging Model for Thick Objects

An imaging model based on a waveguide descriptionof thick objects was described by Nyyssonen4 forvertically edged one-dimensional 11-D2 structuresand later applied to nonvertical edges by NyyssonenandKirk.5 Bothmodels were restricted to TE polar-ization, that is, with an incident electric field oscillat-ing parallel to the wafer structure. More recentlyYuan and Strojwas6 have extended the model toinclude TM or any arbitrary state of polarization andhave reported good agreement with measured differ-ential Nomarski images of shallow objects 1150 nm2.In summary, the two-dimensional 12-D2 waveguide

model assumes that a line object of width W anddepth d forms part of a periodic grating, as shown inFig. 1. For the TE case, an incident electric fieldpolarized in the y direction 1oscillating parallel to thegrating, as shown in the figure2 gives rise to ascattered field that is also polarized in the y direction.For TM illumination, the incident and the scatteredelectric fields are polarized in the plane of incidence,having components in the x and the z directions.TE and TM polarizations are treated quite indepen-dently; the total transmitted or reflected field for anysituation is given by a linear combination of the TEand the TM solutions.The plane wave illuminating the object of Fig. 1

gives rise to a reflected field propagating back intofree space 1region 12 and a transmitted field propagat-ing through the grating 1region 22 into the areaoccupied by the substrate 1region 32. In each of thethree regions the electromagnetic field is expanded

Fig. 1. Object representation in the waveguide imaging model.

132 APPLIED OPTICS @ Vol. 35, No. 1 @ 1 January 1996

in a series of orthogonal solutions or eigenfunctions,each satisfying the TE or the TM wave equationwithin that region. In addition, the eigenfunctionsof region 2 also satisfy the boundary conditionsdefined by the periodic form of the object. We findthe expansion coefficients for each region by invok-ing the boundary conditions at the interface betweenregions 1 and 2 and between regions 2 and 3, namely,that the tangential components of the electric andthemagnetic fields are continuous across any discon-tinuity in relative permittivity. The resulting equa-tions may then be solved numerically with matrix-inversion techniques. The complex amplitude ofthe scattered field in region 1 is used in the conven-tional scalar imaging formulation to calculate thefinal image.Most real objects have dimensions that vary with z

1bars with sloping or rounded edges for example2.Such objects are modeled by slicing region 2 intoseveral layers5 whose values of d, W, and L areconstant over a finite interval in z. The extent ofeach slice will depend on the slope and the width ofthe object at any given z position. The waveguidemodes are solved for each layer and then equatedwith adjacent layers, so that the fields at z 5 0 andz 5 dmay be linked by a series of matrix operations.

3. Linnik Interference Microscope

A. Optical Layout

The primary measurement system constructed toprovide experimental images is a computer-con-trolled phase-stepping Linnik interference micro-scope, shown schematically in Fig. 2. Bright-fieldillumination is provided by either a 100-W tungsten–

Fig. 2. Optical schematic of the automated Linnik interferencemicroscope.

halogen lamp or a 3-mW helium–neon laser. Forthe latter, a fast-spinning ground-glass diffuser aver-ages out speckle noise. Lenses L1–L3 ensure Kohlerillumination so that the variable aperture stop isimaged at the back focal plane of the objectives.To facilitate confocal operation,7 a second helium–neon laser can be switched into the system by adichroic mirror. The dichroic mirror allows theconfocal spot and the full field to be viewed simulta-neously in the eyepiece during setting up. A spatialfilter emulates a point source, the light from which iscollimated by lens L3. Because the objective lensesare infinity corrected, the point source is focused atthe sample.The incident illumination is divided at a glass

beam-splitting cube into reference and sample arms.A three-position stop allows either arm to be viewedindependently or with both arms together for inter-ference. Matched objective lenses are located ineach arm, mounted on dovetail slides to permit rapidchange of magnification with accurate realignment.For the research presented here, 0.9 N.A. 1003objectiveswere used throughout.8 These lensesweretested on a Twyman–Green interferometer and foundto have less than l@8 aberration across the pupil.The reference arm terminates at a l@20 mirror withstrips of differing reflectance, one of which is selectedto match the object reflectivity and to maintain goodfringe contrast. Tilted parallel glass plates in eacharm may be rotated manually to adjust the interfer-ence fringe spacing and direction. Coarse or finepairs of opposing glass wedges facilitate manuallycontrolled or computer-controlled path-length adjust-ment, respectively.Beyond the interferometer, a tube lens brings the

collimated imaging rays to an intermediate focus,which may be relayed to one of several bright-fielddetectors by a projection eyepiece or directed onto anaperture in front of a photomultiplier tube for scan-ning modes of operation. A rotatable polarizer andanalyzer are positioned in front of and behind theinterferometer section, respectively. They are usedin conjunction with a l@2 wave plate to select eitherTE or TM polarization with respect to the sample.

B. Mechanical Construction

The majority of the system is secured to a horizontalaluminum plate, forming an optical table. This isfirmly secured to the top of a large box angle platemeasuring approximately 23 cm 3 46 cm 3 30 cmdeep 1Fig. 32. Attached to one side is a secondL-shaped plate 1shown cut away in the figure2 fromwhich the sample stage is suspended. Both of thesesections are made from cast iron and form a massiveframe around which the rest of the microscopehangs. The sources are situated to one side of thebox angle plate and are shielded from it by areflective aluminium baffle 1not shown2. The confo-cal laser source is secured vertically, and the beam isturned into the system before the spatial filterassembly. Bright-field illumination is directed ontothe optical table by way of a periscope that incorpo-

rates the filter holder, aperture stop, and source-selecting mirror. Both bright-field sources are at-tached to the periscope column and do not contactany other part of the instrument.All the reference and sample arm components lie

on the optical table, with the exception of the sampleobjective, which is located vertically above the stage.The sample arm is reflected down through a hole inthe overhanging L plate by a 90° turning prism.An identical prism in the reference arm folds theoptical path while keeping it on the table. Thebright-field detector arm is turned upward so thatthe viewing eyepiece can be located at a convenientworking height.The large box angle plate and periscope column

are firmly bolted to a 3 ft 3 3 ft 191.44 cm 3 91.44 cm2lapped surface table of cast iron with 3-in. 17.62-cm2ribbing on the underside. The table sits on inflatedtire inner tubes sandwiched between sheets of block-board. This gives a reasonable degree of vibrationisolation from the building in which the instrumentis housed. The optical table is enclosed beneath analuminum cover that contains various access hatches.The vertical detector head and illumination peri-scope are enclosed in tubing throughout. The instru-ment is thus shielded from external light, dust, andair turbulence.

C. Sample-Stage Assembly

The hanging sample-stage assembly replaces anearlier arrangement9 in which individual position-ing stages were stacked one above the other on thesurface table. Such an arrangement was found tointroduce unacceptable path-length drifts into thesample arm. The complete stage assembly is illus-trated in Fig. 4, as viewed from the front of themicroscope. Three steel pillars hold a triangularsupport table at its apices approximately 150 mm

Fig. 3. Physical layout of the microscope 1see Fig. 2 for key2.


below the angle plate. The pillars, shown cutawayfor clarity in the figure, can be extended with match-ing spacers to accommodate deeper samples. Threemicrometers push against a second triangular plateto provide a tilt stage for leveling and coarse focusingof the sample. Tension springs between the platescomplete the kinematic arrangement.A 1-D object-scanning stage uses strip hinges

arranged as parallelograms to provide accurate,frictionless movement of the sample 1perpendicularto the plane of the figure2 while minimizing move-ment in the z direction. The stage is piezoelectri-cally driven under computer control and incorpo-rates position feedback to minimize hysteresis.The sample is positioned on a circular vacuum chucksecured to the stage. A series of concentric groovesmay be evacuated to pull the sample down, makingthe task of locating features easier 1this is done bymoving the sample by hand2 and ensuring interfero-metric stability. The stage assembly is enclosed bya transparent Perspex cover between the surfacetable and the overhanging angle plate.

D. Computer Control and Operation

Phase-shifting interferometry10 1PSI2 permits rapidretrieval of phase and intensity images in the micro-scope. We use Carre’s method of recording fourinterferograms with a phase step of approximately90° between each image.11 Apiezoelectrically drivenvariable-thickness glass block in the sample armintroduces the phase steps, and interferograms arerecorded with fluffed fringes. The glass block con-sists of two opposing wedges with a wedge angle of0.5°. Fine control of focus is effected by moving thesample objective along the optic axis. A piezoelec-tric translator mounted in a strip-hinge assemblydrives the objective and ensures smooth repeatablefocusing.Apersonal computer 1PC2 controls the piezoelectric

devices in the microscope and collects and processes

Fig. 4. Microscope sample stage.


the 1-D intensity data. The control and processingsystem is shown in Fig. 5. Bright-field images arerecorded by a 384 3 491 pixel CCD camera andstored on an 8-bit video digitizer board. For scannedimages, a photomultiplier tube is read out by ananalog–digital converter. The object-sampling inter-val is determined by the ratio of image-sampling rateto stage scan speed; sampling intervals of between12 and 50 pixels per micrometer are available, withcorresponding scan lengths of between 40 and 10µm. Each scanned profile consists of 500 12-bitintensity values.Data input and control output are handled by

commercial software subroutines. Custom assem-bly language subroutines handle all data processingoperations, including phase calculation, phase un-wrapping, and display. An 800-line BASIC programmanages the subroutines and provides user inter-face 1for setting the sampling interval, scan averag-ing parameters, etc.2. 1-D phase and intensity pro-files derived from 100 averaged video lines can beacquired, processed, and displayed in 6 s. Photomul-tiplier images are slower because of a scan rate of ,1Hz 1a value chosen arbitrarily2. The acquisition,processing, and display time for intensity and phaseprofiles derived from nine averaged scans per phasestep is ,70 s.

4. Microscope Performance Characterization

Microscope performance is summarized in Table 1.Individual performance characteristics are consid-ered below.

A. Sampling Interval and Focus

Lateralmagnificationwas calibrated by themeasure-ment of distances between similar image features ona photomask linewidth standard. In the scanningmode, measurements were taken at different regionsof the scan to check the linearity of the driving piezounder load. With position feedback, the stageshowed a linear response to within the accuracy ofmeasurement. The fine-focus mechanism was cali-brated in situ against a portable length-measuringinterferometer. The mechanism was found to show

Fig. 5. Microscope control and processing system: PMT, photo-multiplier tube; A@D, analog–digital; PZT, piezoelectric trans-ducer.

a nonlinearity of ,1% that was due to piezohyster-esis.

B. Optical Resolution

A chromium line object of width below the resolutionlimit of the microscope was used to determine theline-spread function for each scanningmode of opera-tion. Following the definitions used in Ref. 7, theseare 1i2 bright-field scanning type 11a2 1an extendedsource provides Kohler illumination at the object,which is scanned beneath a point detector2, 1ii2 scan-ning probe type 11b2 1a point source illuminates theobject, which is scanned beneath a large-area detec-tor2, and 1iii2 confocal scanning type 2 1a point sourceilluminates the object, which is scanned beneath apoint detector2.The measured responses are shown in Fig. 6 after

the removal of the dc background and centering–normalizing by the fitting of Gaussians to eachresponse. The dc background was produced by theglass substrate supporting the line object and ob-scures any detail beyond the central maximum.The dc asymmetry in the images results from asym-metry in the object, whereas the small peak occur-ring at ,0.6 µm indicates on-axis aberrations, whichmost likely arise from assembly errors in the objec-tive. This is verified by conducting similar scanswith the object reversed. The confocal half-widthshows an improvement of ,10% over that of type11b2. Although in theory type 11a2 and type 11b2 sys-tems are equivalent, their responses differ becauseof the finite source and detector apertures used.Measurements of the full width at half-maximum

Table 1. Microscope Characteristics a

Character-istic Specifications

Wavelength 632.8-nm monochromaticObjectivelenses

1003, 0.9 N.A. achromats

Coherence Continuously variable; 0.1 , S , 1.0Object sam-pling Mode

SamplingInterval

ScanLength

1nm2 1µm2Bright-field 67.1 6 0.8 24.42 6 0.30Scan low res. 78.6 6 1.0 37.92 6 0.48Scan high res. 19.5 6 0.3 9.75 6 0.12

Fine focus Range 10.7 µm in steps of 10 nmOptical reso- 0.45 µm 3type 11a2 scanning4lution 0.38 µm 1confocal scanning2

Phase pro-filing

Carre PSI technique: four steps of 90°

Phase reso-lution

1°, limited by PSI algorithm lookup table

Accuracy 60.47° 1l@7602 1scanning modeb2Repeat-ability

60.48° 1l@7502 1scanning modeb2

Processing ,6 s 1bright-field modec2speed ,115 s 1scanning modeb2

aSee Section 4 for details.bAveraging 16 stage scans per phase step.cAveraging 100 video lines per phase step.

were used to calculate the experimental resolution ofthe microscope 1see Table 12, assuming the conven-tional form of the line-spread function 3I1x2 5 sin1x22@x24in each case.

C. Numerical-Aperture Factors

It is well known that the measurement of phase in ahigh-N.A. interference microscope is subject to acorrection factor because of the difference betweeneffective and nominal N.A.’s.12 The bright-field illu-minating aperture in our microscope was adjustablebymeans of the aperture-stop iris diaphragm 1Fig. 22;hence the variable effective N.A. required calibrat-ing at each stop setting. Phase-correction factorswere calculated at each stop position by themeasure-ment of the height of a known step height standard.The values obtained were too small, being less thanunity for low apertures 1N.A. factors are always .1in reflection microscopes2. The coherence param-eter S at each setting was calculated from the ratio ofsource diameter in the objective back focal plane toobjective pupil diameter. Values of S were thenused to calculate the effective N.A. according to

S 5N.A.effectiveN.A.imaging

, 112

where N.A.imaging is the nominal N.A. of the objective.Plotting N.A. factor versus effective N.A. for eachstop setting and comparing the results with theoryat low apertures 1see Ref. 122 enabled corrections tothe N.A. factors to be made. Values of the coher-ence parameter were used in the waveguide-model-ing process.

D. Accuracy of Phase Measurements

Careful consideration was given to the eliminationor the reduction of sources of error in the calculatedphase returned by the microscope. Simulations ofphase-shifting errors showed a mean error in thecalculated phase of 60.32° resulting from a 61°

Fig. 6. Experimental line responses for different scanningmodesin the microscope. Profiles of a chrome line on glass.


accuracy in the setting of the phase steps. Thesystem wave front was investigated by profiling al@20 aluminized flat placed on the sample stage.The most notable feature of the system phase profilewas a repeatable position-dependent error of 62.5°introduced by the scanning stage over the central 10µm of displacement. This scan length correspondsto the smallest sampling interval, used for theresults presented here, and indicates a stage displace-ment of 62.2 nm parallel to the optic axis. Thesystem profile was stored in a lookup table andsubtracted from measured profiles. The effect ofsignal noise on the calculated phase was simulated,and results were obtained for typical measuredvalues of rms intensity. The error was reduced to60.15° by the averaging of 16 scans at each phasestep. Simulations of quantization noise that wasdue to analog–digital conversion showed errors of60.14° for the CCD detector and less than 60.01° forthe photomultiplier. The use of a lookup table inthe PSI algorithm to convert arctangent values led toa more serious phase-quantization error of 60.29°.Random errors introduced by vibration, air turbu-lence, and thermal drift were minimized by therecording of data during quiet periods on weekendsor overnight. In summary, it was found that, when16 scanned intensity profiles were averaged perphase step, the total error in the calculated phasecould be kept to within 60.47°.

E. Repeatability of Phase Measurements

Measurement repeatability may be investigated bytakingmany phasemeasurements for a given sampleand determining the standard deviation at eachpixel. Approximately 70 phase profiles were re-corded with the CCD camera in ,11 min, duringwhich time the measured phase drift that was due totemperature variations was less than 1°. Bright-field repeatability 1averaging 100 video lines perphase step2was60.52°. For scanning operation thelonger data collection time leads to unacceptablylarge phase drifts. Assuming that a Gaussian errorsource determines repeatability, we note that thestandard deviation of the measured phase at a singlepixel, taken over many measurements, is equal tothe standard deviation of the phase difference be-tween two data sets, taken over all pixels.13 Thusthe measurement repeatability for the scanningmode is obtained by calculating the rms phasedifference between two consecutive phase profiles.The mean of 10 such measurements gave a repeat-ability for the scanning mode 1each phase step aver-aging 16 scans2 of 60.48°.

5. Profiles of a Thick Wafer Object

A. Wafer Object

The test sample is a silicon wafer overcoated with anoxide layer of 666-nm nominal thickness. Sets ofparallel grooves etched into the oxide layer revealthe silicon substrate beneath. The grooves formtest sites of approximately 250 µm 3 250 µm 1Fig. 72,


repeated across the surface of thewafer. Thewidthsof the grooves vary nominally from 0.4 to 2.5 µm in0.1-µm intervals, although in practice grooves below0.7 µm are of poor quality or nonexistent. Opticalmeasurements were made across the center of thewidest set of grooves in the extreme top right-handcorner of a single test site. The quantities underinvestigation are the phase of the interferogram inthe Linnik microscope and the square of the productof the moduli of the complex amplitudes of the twobeams, as returned by the PSI algorithm. Thesequantities are referred to below and in the figures asphase and intensity.

B. Experimental Profiles

Experimental profiles were recorded by using twomodes of microscope operation, both employing ob-ject scanning. For bright-field 3type 11a24 scanning,Kohler illumination was effected at the sample, and

Fig. 7. Test site on a wafer sample. Numbers indicate thenominal width in micrometers of etched grooves.

Fig. 8. Raw phase and intensity profiles from the Linnik micro-scope; silicon oxide bar on silicon, nominal bar height 666 nm.

a slit of width 6.7 µm was placed in front of thephotomultiplier detector. The slit was oriented par-allel to the wafer grooves and gave a 67-nm-widefootprint at the object. The length of the slit was setby the field stop at ,5 µm in the object plane. Thebright-field coherence parameter S was 0.3, givingan effective N.A. of 0.27 162%2. The N.A. correctionfactor at this stop setting was 1.02. For confocalscanning, an 8.5-µm-diameter source pinhole wasimaged at the sample and a 10-µm-diameter detectorpinhole was placed in front of the photomultiplier;projected footprints at the object plane were 0.13 and0.1 µm, respectively. Both scanning modes usedlaser light sources at l 5 632.8 nm.

Fig. 9. Physical parameters and refractive indices available formodeling the wafer sample.

The highest sampling interval of 19.5 6 0.3 nmwas used for both operating modes, giving a scanlength of 9.75 µm with 500 data points. Sixteen in-tensity scans were averaged for each phase step inthe PSI routine; typical PSI output is shown in Fig. 8.Sets of phase and intensity profiles were calculatedfor a 1.5-µm range of focal positions, taken at 0.05 60.01-µm intervals of defocus. Separate sets wererecorded for TE and TM polarization in each operat-ing mode.Profile sets were transferred from PC hard disk to

a Sun workstation for processing. It can be seenfrom Fig. 8 that each profile includes just under twocomplete periods of the bar@groove structure. Prepa-ration of these raw profiles was as follows:

1i2 Removal of a small lateral displacement of ,5nm between consecutive profiles 1presumably causedby residual piezocreep2.1ii2 Equal shift of all profiles in x so that the origin

lies at the center of a chosen bar.1iii2 Windowing of profiles to display a single

sample period; 22.5 µm # x # 12.5 µm.1iv2 Centroiding of each phase profile to give a

mean phase of 0° across the window.

C. Theoretical Profiles

Object parameters available for use in the wave-guide model are shown in Fig. 9. Electron micro-graphs of the wafer edge cleaved along the widestgrooves suggest slopes of ,70°, resulting in a bar-to-

Fig. 10. Bright-field intensity and phase profiles of a centered 1-D bar object 1silicon oxide on silicon2 with vertical edges. Nominal barheight: 1a2, 1b2 151 nm; 1c2, 1d2 640 nm. Comparison of scalar and waveguide images: ——scalar, - - - - - -waveguide TE, ––––waveguideTM. All profiles are focused at the top of the bar.


1a2

Fig. 111a2. Bright-field intensity and phase profiles of a centered 1-D bar object 1silicon oxide on silicon2: Measured bar height 640 nm,——experimental profile, - - - - - -theoretical profile. Profiles are at four focal positions 1a2–1d2 measured from the top of the bar; TEpolarization throughout. This sequence is continued in Fig. 111c2.


1b2

Fig. 111b2. Confocal intensity and phase profiles of a centered 1-D bar object 1silicon oxide on silicon2: measured bar height 640 nm,——experimental profile, - - - - - -theoretical profile. Profiles are at four focal positions 1a2–1d2 measured from the top of the bar; TEpolarization throughout. This sequence is continued in Fig. 111d2.


1c2

Fig. 111c2. Bright-field intensity and phase profiles of a centered 1-D bar object 1silicon oxide on silicon2: measured bar height 640 nm,——experimental profile, - - - - - -theoretical profile. Profiles are at four focal positions 1e2–1h2 measured from the top of the bar; TEpolarization throughout. This sequence is continued in Fig. 111e2.


1d2

Fig. 111d2. Confocal intensity and phase profiles of a centered 1-D bar object 1silicon oxide on silicon2: measured bar height 640 nm,——experimental profile, - - - - - -theoretical profile. Profiles are at four focal positions 1e2–1h2 measured from the top of the bar; TEpolarization throughout. This sequence is continued in Fig. 111f 2.


1e2

Fig. 111e2. Bright-field intensity and phase profiles of a centered 1-D bar object 1silicon oxide on silicon2: measured bar height 640 nm,——experimental profile, - - - - - -theoretical profile. Profiles are at four focal positions 1i2–1l2 measured from the top of the bar; TEpolarization throughout.


1f2

Fig. 111f 2. Confocal intensity and phase profiles of a centered 1-D bar object 1silicon oxide on silicon2: measured bar height 640 nm,——experimental profile, - - - - - -theoretical profile. Profiles are at four focal positions 1i2–1l2 measured from the top of the bar; TEpolarization throughout.


space ratio of approximately 1:1.3 at midheight.Measurements of the groove height and period weremade with a Talystep stylus profilometer and theLinnikmicroscope, respectively. Repeated Talystepmeasurements gave a step height of 640 6 4 nm.Microscope parameters used for modeling includedillumination at 632.8 nm, an N.A. correction factor of1.02, and an imaging N.A. of 0.9. These valueswere also used for the theoretical simulations ofFigs. 10, 12, and 13 below. Sets of theoreticalprofiles were computed at 0.1-µm intervals of defo-cus 1twice the experimental interval2 with programsrunning on a Sun workstation. Because of con-straints of program computing time, the object wasmodeled with vertical edges instead of the apparent70° slope. This removed the need to calculate a setof waveguide equations corresponding to discretevertical slices of the object. Bright-field imageswere computed assuming a line source in the backfocal plane of the microscope, as a means of furtherreducing program complexity and computing time.This 2-D approximation was considered to be lessvalid at higher illuminating N.A.’s, so the confocalimages were computed with a more accurate 3-Dapproximation, requiring the calculation of imagesfor all points on a circular source.14 These programsimplifications are discussed further in Section 6.

D. Modeling the Object: Scalar versusWaveguide Theory

Before experimental and theoretical profiles arecompared, it is informative to model our wafer objectby using both the waveguide and the scalar imagingtheories. The latter uses the Fresnel equations tocalculate the phase and the amplitude of the re-flected field at the substrate and the thin-film reflec-tion formulas for the field at the bar. Because themodel uses scalar waves, the polarization state of theincident and the reflected light is not taken intoaccount. Figure 10 compares bright-field phase andintensity profiles calculated with both models, forthin 1151-nm2 and thick 1640-nm2 vertically edgedsilicon oxide bars on silicon. Focus is at the top ofthe bar, and the intensity is normalized to unity foran object that consists of a perfectly conductingmirror. Results for the thin wafer 3Figs. 101a2 and101b24 show that for shallow objects the conventionalmodel is adequate, although there are visible differ-ences between the scalar and the TE and TMprofiles.For the thick wafer 3Figs. 101c2 and 101d24 the twoapproaches give very different results because thescalar theory fails to consider diffraction effectswithin the object. If one considers the phase differ-ence between light reflected from the oxide bar andfrom the silicon substrate 1approximately 340°2, thenthe scalar approach assumes a discontinuity in thereflected wave front and interprets the difference asbeing only 340° 2 360° 5 220°. In the waveguideapproach there is no discontinuity in the phase of thediffracted field, and so the phase and the intensityimage profiles oscillate heavily in the region of thebar edges. Note that for homogeneous or metal-


lized samples, the phase change f on reflectionwould be a simple function of step height t, where

f 52p

l2t. 122

This gives a phase difference of 172° and 758° forthin and thick steps, respectively, ignoring any N.A.factor. Equation 122 is typically used by opticalinterferometric surface profilingmicroscopes to mea-sure object height.

E. Results

Experimental and theoretical profiles are plottedtogether in Fig. 11. To reduce the quantity of data,only TE images are shown. Figures 111a2 and 111b2show bright-field and confocal images, respectively,beginning at a focal position 0.2 µm above the oxidebar 1the four top images2 and thence focusing downinto the object. The focal sequence is continued inFigs. 111c2 and 111d2, and ends with Figs. 111e2 and 1f 2,displaying a total of 12 focal positions for each mode.We paired experimental and theoretical profiles bysliding the experimental sets through focus to obtaina best match by eye with the theoretical data.Hence only odd- or even-numbered experimentalprofiles are displayed, depending on which set gavethe better match. Each set of experimental inten-sity profiles was normalized with the maximumtheoretically calculated intensity at the substrate.Experimental phase profileswere individually shiftedby a dc term where appropriate. Note that theStrehl limit of defocus gives60.14 µm, which roughlycorresponds to the focal range 1b2–1 j2 in the figures,given the extent of the object.It is seen that agreement between experimental

and theoretical profiles is in general extremely goodfor intensity and phase images in both operatingmodes. For bright-field intensity and phase, thetheory shows tight oscillations at the center of thebar, which are not always consistent with the experi-mental images. The same inconsistency is evidentat the center of the groove 162.5 µm2 in some of theintensity profiles. Conversely, many of the subtlechanges between adjacently focused images havebeen clearly picked up by both the microscope andthe waveguide model 3for example, the bright-fieldintensity lobes at 61 µm, focal positions 1h2–1k2 andthe confocal phase ripples at the bar center, focalpositions 1e2–1h24. The confocal images show lessringing for all profiles, and the intensity images growand diminish rapidly because of the optical section-ing property of the confocal microscope. Here themain discrepancy lies in the contrast of intensityimages. It was found impossible to match the con-trast at more than two or three focal positions, withthe best compromise achieved by normalizing to themaximum intensity at the substrate. The phaseears in the vicinity of the bar edge are narrower forthe confocal profiles, indicating an improvement inresolution, as expected. The brief unwrapping ofthe confocal phase in Fig. 111b2, focal positions 1b2–1d2,

occurs because the extra resolution in the confocalmode allows the phase to resolve the 2p ambiguity atthe bar edge.

6. Choosing and Adjusting the Modeling Parameters

The good agreement shown in Fig. 11 was reachedafter several adjustments to the input parameters ofthe waveguide model. Measurements of the periodL and height D of the bar were considered to bereliable, and refractive indices of bar and substratematerial were taken from standard tables. Theseparameters were thus fixed before modeling. Edgeslope and bar-to-space ratio were less well definedand were considered legitimate variables in themodeling process.The wafer was initially modeled with vertical

edges to check program performance and to obtainsome trial through-focus image sequences. Phaseand intensity profiles for a thin 1,200-nm2 bar arealmost invariant to a change in edge slope from 70°to 90° but the effect is significant for a 640-nm bar, asshown in Fig. 12. The influence of the sloping edgeis best seen in the ears of the phase profiles, whichoccur in the region of the bar edge. As expected, thetransition from vertical to sloping edges gives thephase profile more time 1more resolution2 to followthe physical step, resulting inmore pronounced ears.We found that the vertical edge approximation gavegood agreement between experiment and theory,with the phase ears matching well over a large focalrange. This suggests that the appearance of slop-ing edges in the electron micrographs may be some-

what exaggerated. Following a decision to use ver-tical edges for all modeled sets of profiles, thebar-to-space ratio was adjusted by trial and error toproduce the best agreement with experimental pro-files across a wide focal range. The optimum valueof bar width was 1.975 6 0.025 µm.A second simplification in the modeling process

was to use the 2-D source approximation for bright-field images. Figure 13 compares 2-D and 3-Dmodels for a thin 3Figs. 131a2 and 131b24 and a thick3Figs. 131c2 and 131d24 bar object for which the experi-mental illumination parameters outlined in Section5 were used. Both models are virtually identical forthe thin wafer, but show subtle differences for thethick profiles. Comparisons were made betweenbright-field experimental profiles and the waveguidemodel by incorporating the 3-D source, but no im-provements in the matching of profiles was obtained.

7. Polarization State

Figure 11 shows profiles for TE polarization only.Similar results for TM polarization indicate equallygood agreement, but are not reproduced here be-cause of lack of space. In general the TE profilesexhibit more ringing in both intensity and phase andfor both imaging modes, but the differences aresubtle. This difference becomes more evident onviewing how the profiles change with focal position.Figure 14 shows sequences of overlaid bright-fieldprofiles calculated at multiple focal positions. Pro-files in a sequence are separated by 0.1 µm of defocusthat covers a range of 0.5 µm, 0.6–1.0 µm below the

Fig. 12. 1a2, 1b2 Bright-field TE; 1c2, 1d2 TM intensity and phase profiles produced by the 2-D waveguide imaging theory. Centered 1-D barobject 1silicon oxide on silicon2, height 640 nm. ——vertical edges, - - - - - -edges sloping at 70°. All profiles are focused at the top of the bar.


Fig. 14. Bright-field intensity and phase profiles produced by the 2-D waveguide imaging theory. Centered 1-D bar object 1silicon oxideon silicon2with vertical edges, height 640 nm. Each graph shows an overlay of profiles for focal positions 0.6–1.0 µm below the top of thebar, in 0.1-µm intervals. 1a2, 1b2 TE polarization; 1c2, 1d2 TM polarization.

Fig. 13. Bright-field TE intensity and phase profiles produced by the waveguide imaging theory. Centered 1-D bar object 1silicon oxideon silicon2 with vertical edges. Nominal bar height: 1a2, 1b2 151 nm; 1c2, 1d2 640 nm; ——2-D model, - - - - - -3-D model. All profiles arefocused at the top of the bar.


Fig. 15. Bright-field intensity and phase profiles from the Linnik microscope. Centered 1-D bar object 1silicon oxide on silicon2;measured height 640 nm. Each graph shows an overlay of profiles for focal positions 0.6–1.0 µm below the top of the bar, in 0.1 6

0.01-µm intervals. 1a2, 1b2 TE polarization; 1c2, 1d2 TM polarization.

top of the bar. The sequences of Figs. 141a2 and 141b2are for TE polarization, and the sequences of Figs.141c2 and 141d2 are for TM polarization. The chosenfocal range corresponds to the region of greateststability in the phase image, which is also the regionin which the intensity image changes most rapidly3Figs. 141a2 and 141c24. Note that this region ofdefocus corresponds to a focal plane that is largelylocated within the substrate.The experimental versions of the sequences shown

in Fig. 14 are reproduced for comparison in Fig. 15.The rapidly changing intensity profiles show in-stantly to which polarization state each set of imagesbelong. The nodes that occur at 60.9 µm fromimage center are clearly reproduced in the micro-scope profiles. The TE sequences are generallybetter matched than those for TM polarization. Inthe latter, the spreading of the lobes in the x direc-tion, which gives rise to the thickness of the multipleintensity and phase plots 3Figs. 141c2 and 141d24, is notreproduced experimentally. This is somewhat sur-prising, as themodel assumed vertical bar edges; onemight expect the theoretical profiles to be tighter in xas a result.

8. Summary and Conclusions

Results have been presented for a simple objectconsisting of a silicon oxide bar on silicon, and goodagreement with the waveguide theory has been

obtained for both Type 11a2 bright-field and confocalimages. TE- and TM-polarized illuminations givesimilar profiles but with subtle differences, whichhave been picked up by both the microscope and thetheory.Further research is required for improving agree-

ment between theory and practical images. In par-ticular correct modeling of the microscope requiresadditional investigation. 1Attempts to include theeffect of aberrations by incorporating a measuredvalue of the pupil function into the imaging theorymodified the finer details in the ringing but failed togive any improvement in the matching of data.2The general trend toward increased ringing in thetheoretical bright-field profiles suggests incorrectcalibration of the illuminatingN.A. in themicroscope.Repeating the experimental or theoretical data setsfor different N.A. settings would thus be a usefulexercise. It has been mentioned that implementingthe 3-D source model for a bright-field mode failed toshow any improvement in the matching of profiles.Nevertheless, the subsequent modification of theo-retical images shown in Figs. 131c2 and 131d2 shouldclearly be taken into account in further studies.The stability of the bright-field phase ears over a

defocus range of 0.5 µm enabled the edges of the barto be located to an accuracy of 25 nm, assuming avertically edged structure. This region of stability3and the nodes in the accompanying intensity pro-


files, Figs. 141a2 and 141c24 may be of use for dimen-sional measurement of thick structures, although itis not known within what limits of object width thislevel of accuracy can be applied.We have presented results for an optically thick

symmetrical object with relatively large lateral di-mensions. Thus we have been able to study theperformance of the waveguide model on relativelyisolated object features while taking advantage ofthe anticipated image symmetry to facilitate thematching of experimental and theoretical profiles.The real interest lies in the ability to profile struc-tures with submicrometer dimensions accurately,and it is hoped that, with further improvements inthe validation of modeling and experimental tech-niques, we can begin to study such objects withconfidence.

This research was supported by the U.K. Scienceand Engineering Research Council. Funds for theconstruction of the Linnik microscope and a three-year postgraduate bursary for D. M. Gale wereprovided by IBM, the East Fishkill facility, NewYork. The authors gratefully acknowledge the con-tributions of Fred Reavell and Ted Bates in thedesign and the construction of the Linnik micro-scope, and Mike Downs and colleagues at the Na-tional Physical Laboratory, Teddington, Middlesex,U.K., for assistance with the test samples, electronmicrographs, and general advice. The authors alsothank Mike Barnett for his valuable support anduseful discussions throughout the project.

References and Notes1. W. M. Bullis and D. Nyyssonen, ‘‘Optical linewidth measure-

ments on photomasks and wafers,’’ in VLSI Electronics:Microstructure Science, G. Einspruch, ed. 1Academic, NewYork, 19822, Vol. 3, pp. 301–346.


2. D. Nyyssonen, ‘‘Calibration of optical systems for linewidthmeasurements on wafers,’’ Opt. Eng. 21, 882–887 119822.

3. M. J. Downs and N. P. Turner, ‘‘Application of optical micros-copy to dimensional measurements in microelectronics,’’ inMicroscopy: Techniques and Capabilities, L. R. Baker, ed.,Proc. Soc. Photo-Opt. Instrum. Eng. 368, 82–87 119822.

4. D. Nyyssonen, ‘‘Theory of optical edge detection and imagingof thick layers,’’ J. Opt. Soc. Am. 72, 1425–1436 119822.

5. D. Nyyssonen and C. P. Kirk, ‘‘Optical microscope imaging oflines patterned in thick layers with variable edge geometry:theory,’’ J. Opt. Soc. Am.A 5, 1270–1280 119882.

6. C. M. Yuan and A. J. Strojwas, ‘‘Modeling optical microscopeimages of integrated-circuit structures,’’ J. Opt. Soc. Am. A 8,778–790 119912.

7. T. Wilson and C. Shepard, Theory and Practice of ScanningOptical Microscopy 1Academic, London, 19842, Chaps. 1 and 3.

8. Objective lenses were NPL achromats purchased as matchedpairs from E. Leitz Ltd., London.

9. D. M. Gale, M. I. Pether, and F. C. Reavell, ‘‘Interferencemicroscopy of surface relief structures,’’ inOptical Microlitho-graphic Technology for Integrated Circuit Fabrication andInspection, H. L. Stover and S. Wittekoek, eds., Proc. Soc.Photo-Opt. Instrum. Eng. 811, 40–47 119872.

10. P. Hariharan, ‘‘Quasi heterodyne hologram interferometry,’’Opt. Eng. 24, 632–638 119852.

11. P. Carre, ‘‘Installation et utilisation du comparateur photoelec-trique et interferentiel du Bureau International des Poids etMesures,’’Metrologia 2, 13 119662.

12. K. Creath, ‘‘Calibration of numerical aperture effects ininterferometric microscope objectives,’’ Appl. Opt. 28, 3333–3338 119892.

13. H. Stahl and J. Tome, ‘‘Phase-measuring interferometry:performance characterization and calibration,’’ in OpticalTesting andMetrology II, C. Grover, ed., Proc. Soc. Photo-Opt.Instrum. Eng. 954, 78–87 119882.

14. M. I. Pether, ‘‘Numerical modelling of image formation in theoptical microscope,’’ presented at the First Institute of PhysicsConference on Applied Optics and Opto-Electronics, Notting-ham, England, 17–20 September 1990; in Applied OpticsDigest, J. C. Dainty, ed. 1IOP Publishing, Bristol, U.K., 19902,pp. 191–192. Further work to be published.

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