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Ultramicroscopy 109 (2008) 1–7

Contents lists available at ScienceDirect

Ultramicroscopy

0304-39

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/ultramic

Beam spreading and spatial resolution in thick organic specimens

Jerome K. Hyun a,�, Peter Ercius b, David A. Muller b

a Department of Physics, Cornell University, E13 Clark Hall, Ithaca, NY 14853, USAb Department of Applied and Engineering Physics, Cornell University, E13 Clark Hall, Ithaca, NY 14853, USA

a r t i c l e i n f o

Article history:

Received 17 January 2008

Received in revised form

28 June 2008

Accepted 8 July 2008

PACS:

68.37.Ma

87.57.cf

Keywords:

STEM

Spatial resolution

Thick section

Plural elastic scattering

91/$ - see front matter & 2008 Elsevier B.V. A

016/j.ultramic.2008.07.003

esponding author. Tel.: +1607 339 7061; fax:

ail address: jkh32@cornell.edu (J.K. Hyun).

a b s t r a c t

Tomography using a scanning transmission electron microscope (STEM) offers intriguing possibilities

for the three-dimensional imaging of micron-thick, biological specimens and assemblies of

nanostructures, where the image resolution is potentially limited only by plural elastic scattering in

the sample. A good understanding of the relationship between material thickness and spatial resolution

is required, with particular emphasis on the competition between beam divergence (a geometrical

effect from the converged STEM probe) and beam spreading (an unavoidable broadening due to plural

elastic scattering). We show that beam divergence dominates beam spreading for typical embedding

polymers beyond the 100-nm thickness range and that minimization of this effect leads to enhanced

spatial resolution. The problems are more pronounced in spherical-aberration-corrected instruments

where the depth of field is shorter.

& 2008 Elsevier B.V. All rights reserved.

1. Introduction

Advances in instrumentation and technique have increased theinterest in electron tomography for biology [1,2] and materialsscience [3]. For problems involving polymers and composites [4],organic low-K dielectrics, photonic bandgap materials [5] andintegrated circuits [6], tomography continues to show promise inproviding quantitative information previously unattainable bytwo-dimensional (2D) electron microscopy.

The optimum mode for electron tomography, TEM or scanningtransmission electron microscope (STEM), for investigating thin,biological weak-phase objects, continues to be debated [7].However, for sample thicknesses on the order of a few hundrednanometers, STEM might offer some advantages [8] since thereneed not be a post-specimen imaging lens and hence nochromatic blurring from energy losses in the sample. For the caseof TEM, an increase in energy spread in thick samples results in adecrease in resolution due to chromatic aberrations. Inclusion ofan energy filter can improve the resolution, but this also reducesthe recorded intensity significantly [9,10].

Electron tomography requires the acquisition of a series of 2Dprojection images tilted in small increments, from which a three-dimensional (3D) object can be digitally reconstructed. In TEM

ll rights reserved.

+1607 255 7658.

mode, diffraction artifacts at different tilts may not be straightfor-ward to interpret and phase contrast effects may requiremathematical post-processing [11]. STEM using a high-angleannular dark field detector (HAADF) [12] minimizes this problemsince the intensity in each image pixel is the sum of elasticallyscattered electrons over an annular range that excludes the Bragg-scattered electrons that are responsible for generating diffractioncontrast.

Ideally the spatial resolution should remain constant, inde-pendent of depth into the sample. However, beam spreading fromplural elastic scattering of the electrons within the sample [10]and beam divergence due to the geometry of a convergent beamboth challenge this condition. In STEM mode, beam spreadingdetrimentally affects image resolution for features located at theexit (bottom) surface, while features located at the entrance (top)surface are imaged with the resolution determined by thediameter of the point spread function. Consequently, top featuresare more reliably reproduced in a STEM image compared tobottom features, a result also known as the top–bottom effect[13]. In TEM mode, the reverse is true [14], where resolution ofbottom features are influenced largely by just the chromaticerrors and top features are influenced by both plural scatteringand chromatic aberration [15].

Since beam spreading depends on material properties and thekinetic energy of the electrons, it cannot be modified withoutchanging the beam voltage. Beam divergence, on the other hand,can be minimized by decreasing the convergence angle of the

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J.K. Hyun et al. / Ultramicroscopy 109 (2008) 1–72

microscope. Often, the effect of beam divergence is under-estimated and beam spreading is thought to set the resolutionlimit.

Earlier measurements by Beorchia et al. [16] illustrate therelation between spatial resolution and thickness for polymersubstrates. With the advances in brightness that commercial FEGelectron sources offer, an analysis of the problem with particularemphasis on the effects of beam divergence and beam spreadingis essential. We demonstrate that for typical convergence anglesfor high-resolution atomic imaging, the dominant effect is thebeam divergence in polymer films thinner than 1.1mm. Tilting thesample increases the projected thickness, thereby enhancingthe effects of beam spreading and beam divergence. In suchcases, a tilt series of a thick specimen could fail to produce areliable tomographic reconstruction. By simply selecting theproper convergence angle, we can optimize the spatial resolutionand improve the accuracy of the reconstruction.

2. Experimental

2.1. Instrumentation

A 200-kV FEI Tecnai F20 STEM system was used for allmeasurements. The optimal convergence angle for high-resolu-tion STEM imaging set by the 1.2-mm spherical aberrationcoefficient is 9.6 mrad [17], where the resulting resolution is1.6 A and the depth of field is 22 nm. The actual convergence angleduring normal operation was measured to be 1070.1 mrad.

The instrument has only two condenser lenses so source sizeand convergence angle cannot be varied independently as in athree-condenser arrangement. Either an additional lens is neces-sary, or a physical aperture must also be changed. Once thesmallest probe-forming aperture is inserted, the convergenceangle can be further reduced by decreasing the objective lensstrength and re-focusing using the final condenser lens (C2).A quicker way is to take advantage of the minicondenser lens,which controls the switching between TEM and STEM mode incombined TEM/STEM systems. By reversing the minicondenserlens current while operating in STEM mode, the beam becomesmore parallel, producing a substantially narrower convergenceangle. We were able to reduce the convergence angle to270.2 mrad using this simple technique. The appropriate adjust-ments in the optical alignment were saved, thereby allowing fastswitching between the 10 and 2 mrad convergence angle settings.This also allows the objective lens strength optimally set foreucentric focus to be preserved, and minimizes drift in the opticalalignments. An alternative technique for significantly reducing theconvergence angle is through the Microprobe STEM mode [18].

The convergence angle a can be determined on a crystallinesample by

a

b¼

ayb

(1)

where a is the aperture diameter, b is the spacing between Braggdiscs, and yb is the Bragg angle for a particular reflection [17]. Allconvergence angles were measured by recording the CBEDpatterns of Si oriented onto the (110) axis and using the (111)reflection for yb.

2.2. Methodology

Various polymer film thicknesses up to 1mm were prepared byultramicrotoming Quetol 651, a widely used embedding mediumfor biological samples [19]. In order to characterize the spreadingand divergence of the beam, gold nanoparticles were deposited on

top and bottom of the films, similar to a method implemented byBeorchia et al. [16]. On the bottom side of the film, an additional30 A of amorphous carbon was sputtered to avoid charging and toincrease thermal stability. The areas of interest were additionallyflooded with electrons in TEM mode for roughly 20 min to allowfor the film to undergo any shrinkage and reach a stable thickness.

The original size of the nanoparticles, determined by measur-ing the full width at half maximum (FWHM) of top particles infocus, was 6.470.4 nm. The average particle diameter wasconsistent for all regions used for the analysis and variationsstayed within the error.

The physical thicknesses of the films were measured by t/li

EELS measurements [20] for thicknesses less than 600 nm. Beyondthis sample thickness, insufficient signal enters the detector tomake reliable measurements. Instead parallax measurementswere used to obtain the thickness for films thicker than 600 nm.

The inelastic mean free path, li, was calibrated to a sample ofknown thickness by acquiring a full tilt series of elastic imagesand using electron tomography to reconstruct a full 3D imagefrom which the sample thickness was determined. From EELSmeasurements on the same sample, the inelastic mean free pathfor Quetol 651 was found to be 110710 nm.

Beam spreading can be defined in several ways. The 90% beamradius is a popular choice as it can be described by a simpleanalytical model, and scales as the 3/2 power of the thickness [21].It is a useful measure for trace-element microanalysis but lesspredictive for imaging because it is dominated by the far tailsfrom infrequent large-angle scattering events, which are often notdetectable at the lower signal to noise ratios used in imaging. Amore appropriate measure for evaluating image resolution is theFWHM of the beam spread. The FWHM has a very differentthickness dependence to the 90% radius and shows a much lesspronounced spreading with sample thickness than does 90%radius [6].

The effects of beam spreading and beam divergence can beseparated by imaging features at the entrance and exit of thesample. Figs. 1a and b illustrate the method of isolating andcharacterizing beam spreading and divergence, respectively. TheFWHM of the bottom particle in focus is affected predominantlyby plural scattering, and hence measures beam spreading.Likewise, keeping the bottom particle still in focus, the FWHMof a top particle is affected only by the defocused beam as there isno material above for plural scattering to occur, and hencemeasures beam divergence.

Assuming that the geometrical divergence, the diffraction limitand the original particle size are Gaussians, we can approximatethe FWHM of the top particle, xtop, by adding these three differentcontributions in quadrature:

xtop ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2t tan aÞ2 þ 0:61l

a

� �2

þ x20

s(2)

where t is the thickness of the sample, a is the convergence angle,l is the electron wavelength and x0 is the FWHM of the originalparticle size.

Calculation of the FWHM of the bottom particle was performedby a Monte Carlo algorithm [22] .The algorithm simulates ascanning electron beam that enters perpendicular to the film. Foreach scan position the elastically scattered trajectories of theelectron at the exit surface are used to determine their paths tothe post-specimen detectors. Electrons scattered into an annularrange corresponding to the physical dimensions of the HAADF arethen collected to generate an intensity profile of a gold particlesitting on the bottom of the film. The FWHM of the profile isthen extracted and compared to that of the experiment. Anelastic mean free path, le, of 360 nm was used, approximately

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Electronbeam

Method A

Polymersubstrate

Amorphouscarbon (~30 Å)

Goldnanoparticle(6.4 ± 0.4 nm)

Method B

30Measured Bottom Particle FWHM (Method A)

Calculated Bottom Particle FWHM (Monte Carlo)

Measured Top Particle FWHM (Method B)Calculated Top Particle FWHM (See Eq. 2)

20

25

10

15

5

10

15

App

aren

t Par

ticle

Siz

e (n

m)

50

Thickness (nm)200 400 600 800 1000 1200

Fig. 1. (a) and (b) Method for characterizing beam spreading and beam divergence. (a) Focusing on the bottom particle and measuring its FWHM provide a measure of the

beam spreading. (b) With the bottom particle still in focus, measuring the FWHM of a top particle provides a measure of the beam divergence. (c) and (d) With bottom

particle in focus, measured FWHM of top (solid circles) and bottom particles (open squares), and the calculated FWHM of top particle using Eq. (2) (solid curve) and bottom

particle using Monte Carlo simulation (dotted curve). (c) A 10-mrad convergence angle setting shows beam divergence dominating beam spreading. (d) A 2-mrad

convergence angle setting shows beam divergence significantly reduced. Original particle sizes are 6.470.4 nm.

J.K. Hyun et al. / Ultramicroscopy 109 (2008) 1–7 3

determined from the inelastic mean free path using

le � 2li 1nð2=yEÞ=Z (3)

where Z is the atomic number and yE describes the characteristicangle corresponding to the mean energy loss [20]. The expressionhas an accuracy of �20%.

3. Results and discussion

Measured and calculated particle size due to beam spreadingand divergence are summarized in Figs. 1c and d. For allmeasurements the bottom particles were kept in focus. The beamspreading effect determines the FWHM of the bottom particlesand the beam divergence effect determines the FWHM of the topparticles.

For a 10-mrad convergence angle (Fig. 1c) the beam divergenceeffect dominates the beam spreading effect and sets the resolutionlimit for measurements up to a micron in thickness. Decreasingthe convergence angle to 2 mrad as shown in Fig. 1d reduces beamdivergence; so the effect becomes comparable to the beamspreading. The depth of field, T, is defined by

T ¼d

a¼

0:61le

a2(4)

where d is the probe size or the diffraction limit, which increases

from 15 to 383 nm by the change in convergence angle from 10 to2 mrad. As a trade off, the STEM resolution effectively defined bythe probe size degrades from 1.5 to 7.6 A from the increaseddiffraction blur. A 1-mrad convergence angle would offer a depthof field of 1.5mm at a cost of a 1.5-nm probe size.

For the 2-mrad setting, the difference between top and bottomparticles is less pronounced than for the 10-mrad setting. Thediscrepancy between experiment and theory in Fig. 1d could beaccounted by the 20% systematic error of Eq. (3), resulting in apossible overestimate of the elastic mean free path used in theMonte Carlo calculation.

The divergence can also be shown in terms of a defocus seriesof the point spread function. Fig. 2 shows the quantummechanically calculated defocus series of a 20-mrad convergenceangle for an aberration corrected system using C3 ¼ �8mm andC5 ¼ 20 mm [23], and that of an uncorrected 10- and 2-mradconvergence angle with C3 ¼ 1.2 mm. With a 20- or 10-mradconvergence angle and defocus values on the order of a 100 nm,the intensity profile of the probe is no longer concentrated in thecenter, but toward the outer tails, leading to a donut or ring shape.This defocused point spread function creates artifacts such as thedeformation of solid particles into rings as shown in Fig. 3. Byfocusing on features buried a few hundred nanometers into thesample, features at the sample entrance surface are no longeraccurate representations, but artifacts of the ring-shaped probeconvolved with the original features. At 2 mrad, the point spread

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300

400

500

600

700D

efoc

us (n

m)

0

100

200

−300

−200

−100

Radius (nm)−4 −2 0 2 4

Radius (nm)−4 −2 0 2 4

Radius (nm)−4 −2 0 2 4

10 mrad 2 mrad20 mrad

Fig. 2. Calculated defocus series of the ADF-STEM point spread function at 200 keV for 20-, 10- and 2-mrad convergence angles, respectively. The 20-mrad convergence

angle setting is calculated for an aberration corrected system, where C3 and C5 values are �8mm and 20 mm, respectively. The 10- and 2-mrad convergence angle settings

are calculated for a C3 ¼ 1.2 mm system. For 20- and 10-mrad convergence angles, the intensity is distributed in the tails of the point spread function that diverges linearly

over the defocus range. For a 2-mrad convergence angle, the intensity is confined within a �1-nm diameter throughout the defocus range.

Fig. 3. Defocus artifacts from a 10-mrad convergence angle. (a) Top particles in focus. (b) The same particles imaged with a defocus of �635 nm, displaying ring-shaped

blurring due to the out-of-focus point spread function.

J.K. Hyun et al. / Ultramicroscopy 109 (2008) 1–74

function stays consistent through its depth of field. Originalfeatures are safely preserved in this range, and are only subject tobeam spreading.

When features near the top of the sample are focused, featuresnear the bottom of the sample are imaged with the convolvedcontributions of beam spreading and divergence.

Fig. 4 displays a comparison between images recorded with a10 mrad and a 2 mrad probe-forming aperture with the beamfocused on top particles sitting on a 600-nm thick film. Thebottom particles appear as a faint haze using a 10-mrad setting,but by decreasing the angle to 2 mrad they come into focus. Thedepth of field at 2 mrad is less than the thickness of the film,which accounts for a slight geometrical contribution to theresolution near the exit side of the film in addition to the beamspreading manifested in the small increase of the FWHM of thebottom particle. The same camera length of 100 mm was used forboth convergence angles. The lower intensity in the 2 mrad settingis due to the different electron optics of the illumination system.

As a practical demonstration, HeLa cells stained withosmium and uranyl acetate were imaged using 10- and 2-mrad

convergence angles as shown in Fig. 5. The cells were embeddedin 400-nm thick polymer film. A camera length of 100 and490 mm was used for the 10- and 2-mrad settings, respectively,resulting in roughly twice more signal in the 2 mrad image.However, the noise levels were 6.1% of the signal for the 10-mradangle and 5.6% for the 2-mrad angle, permitting a fair comparison.Quick inspection of the area in Fig. 5a shows that the filmthickness is non-uniform over the field of view, with a depressionin the top-right region. Fig. 5b shows that both convergence anglesettings identically reproduce a common feature near the bottomof the image. In the depressed area near the top-right region,Fig. 5c shows the wall of an organelle imaged with better clarityand sharpness for the 2-mrad setting than for the 10-mradsetting. Such a demonstration elucidates the consequences ofa tomographic tilt series where beam spreading and beamdivergence increases with the projected thickness.

The eucentric focus is commonly located slightly below themidpoint of the film’s depth, effectively halving the impact ofbeam divergence. However, the typical depth of field is �20 nm inan uncorrected instrument and �5–10 nm in an aberration

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Particle on top

Particle on bottom

6.4 ± 0.6 nm 6.4 ± 0.6 nm

7.2 ± 0.6 nm

14.7 ± 0.6 nm

1.5x104

1.4x104

1.3x104

1.2x104

1.1x104

1.02x104

1.04x104

1.06x104

1.08x104

1.1x104

1.12x104

1.14x104

1.16x104

1.18x104

Cou

nts

Cou

nts

5.5x104

5x104

4.5x104

4x104

3.5x104

3x104

3.4x104

3.35x104

3.3x104

3.25x104

3.2x104

3.15x104

3.1x104

3.05x104

Cou

nts

Cou

nts

mm

mm0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35

mm

0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35mm

Fig. 4. Top and bottom particles on a 600-nm thick film imaged with 10- and 2-mrad convergence angles. Top particles are in focus for both images. (a) The 10-mrad setting

shows the influence of beam spreading and divergence on the bottom particles as they are barely visible. (b) The 2-mrad setting shows a sharper image of the bottom

particles as beam divergence has been reduced. A 100-mm camera length was used for both convergence angles. Different illumination optics results in a lower intensity for

the 2-mrad image.

J.K. Hyun et al. / Ultramicroscopy 109 (2008) 1–7 5

corrected system, so both distances (even after focusing on themidpoint) are much less than a typical 70 to 100 nm-thickmicrotomed section. Furthermore, at high tilts, where theeffective thickness can more than triple, features at the top aresubject to beam divergence and features on the bottom aresubject to both beam divergence and beam spreading. Whenreconstructing a tilt series, blurring of features at large tilt anglescorresponds to a damping of high spatial frequencies in thecorresponding Fourier slice. This results in a loss of reconstructedresolution, most noticeably in the depth direction [24]. In samplesthinner than the depth of field, the effect of beam divergence onthe reconstructed resolution is minimal. However, in samplesthicker than the depth of field, minimizing the beam divergencecan improve the reconstructed resolution.

Although beam spreading cannot be removed, limiting the 3Dsize of a sample by means of advanced sample preparationtechniques such as the focused ion beam can restrict the projected

thickness at large tilts. Since the projected thickness at high tiltsgrows indefinitely only when the length of the sample in thedirection perpendicular to the tilt axis and electron beam is infinite,controlling this length can bound the projected thickness of a tiltedsample. As an example, a typical 0.5-mm thick mesoporous supportwhose width in the direction perpendicular to the tilt axis andelectron beam is 1mm will have a maximum projected thickness ofaround 1.12mm, limiting the effect of beam spreading to thisthickness. Likewise, the effect of beam divergence is also bounded,but can be further reduced by choosing the appropriate convergenceangle, permitting a significant decrease in the blurring of featuresthroughout the depth of the sample.

It is worth noting that the improved spatial resolution in aspherical-aberration-corrected microscope is obtained by increas-ing the convergence angle by a factor of 2–4 over an uncorrectedinstrument, resulting in a decreased depth of field Eq. (4). Intomography, the resolution function d is more likely to be set by

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10 mrad

2 mrad

10 mrad2 mrad

3x1044.4x104

4.2x104

4x104

3.8x104

3.6x104

3.4x104

2.9x104

2.8x104

2.7x104

2.6x104

2.5x104

0 0.04 0.08 0.12 0.16

Arb

itrar

y U

nits

µm

Arb

itrar

y U

nits

0 0.04 0.08 0.12 0.16µm

Fig. 5. Stained HeLa cell imaged with 10- and 2-mrad convergence angles. (a) Image taken with 10-mrad convergence angle. The film is depressed near the top, right region

enclosed by the white box. (b) Expanded view of the bottom boxed region for the 10- and 2-mrad convergence angles. Line profiles across a common feature in both images

show similar profiles. (c) Expanded view of the top boxed region, where features are lower in depth, for the 10- and 2-mrad convergence angles. Line profiles across an

organelle wall show a clearer representation with the 2-mrad setting. All line profiles were aligned vertically by integral.

J.K. Hyun et al. / Ultramicroscopy 109 (2008) 1–76

the accuracy of the alignment in the reconstruction, rather thanthe diffraction limit itself. As a consequence, the reduction indepth of field and corresponding degradation in resolution for acorrected system compared to its uncorrected counterpart is likelyto be by a factor of 2–4 rather than 4–16. In either limit, thecorrector will degrade the resolution for conventional tomographyif the sample thickness exceeds the new depth of field. Instead, totake full advantage of the corrector, it may become necessary torecord a through-focal series at each tilt in order to extend thedepth of field in software. (This should not be confused withdynamic focusing, which records different (x,y) points at differentz settings. Here we require multiple z data for the same (x,y)settings.) The simplest method to process the through-focal seriesis to sum all images in the through-focal series, which can beshown to be mathematically equivalent to recording a projectionimage with an infinite depth of field [23].

4. Conclusion

In very thick (i.e. several microns thick) organic or low-Z materials,beam spreading determines the resolution limit. In moderately thicksamples (0.1–1mm for uncorrected instruments), beam divergence orequivalently the depth of field limits the resolution. In the 0.1–1-mmsample thickness range, the resolution can be optimized by balancingthe geometrical divergence against the diffraction limit. For samplesthinner than the depth of field, the resolution is dominated by thebeam probe size and changing the convergence angle is unnecessary.With the proper adjustment of the optical geometry we can improvethe spatial resolution in thick samples, a method especially importantfor electron tomography. The increased convergence angle inaberration-corrected systems will extend depth-of-field blurring tothinner samples, down to �30–50-nm thick, requiring additionalcollection and processing steps.

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Acknowledgements

The authors would like to thank John Grazul for sharing hisexpertise in biological sample preparation, William Brown forpreparing and providing stained HeLa cell samples, MatthewWeyland for useful discussions, NSF DMR-0405195, the Semi-conductor Research Corporation and facilities MRSEC DMR-0520404 for funding.

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