Making sense of nanocrystal lattice fringes

0.37 MB·English

by  P. Fraundorf

Making sense of nanocrystal lattice fringes
P. Fraundorf
Dept. of Physics & Astronomy and Center for Molecular Electronics, U. Missouri-StL, St. Louis MO 63121 USA∗
Wentao Qin
5 Technology Solutions, Freescale Semiconductor, Inc., MD CH305, Chandler AZ 85284 USA
0
0 Peter Moeck
2 Dept. of Physics, Portland State University, P. O. Box 751, Portland OR 97207-0751 USA

n
a Eric Mandell
J Dept. of Physics and Astronomy and Center for Molecular Electronics, U. Missouri-StL, St. Louis MO 63121 USA

1 (Dated: February 1, 2008)
3
Theorientation-dependenceofthin-crystallatticefringescanbegracefullyquantifiedusingfringe-
] visibilitymaps,adirect-spaceanalogofKikuchimaps. Asinnavigationofreciprocalspacewiththe
i aidofKikuchilines,fringe-visibilitymapsfacilitateacquisitionof3Dcrystallographicinformationin
c
s latticeimages. Inparticular,thesemapscanhelpresearcherstodeterminethe3Dlatticeparameters
- ofindividualnano-crystals,to“fringefingerprint”collectionsofrandomly-orientedparticles,andto
l
r measure local specimen-thickness with only modest tilt. Since the number of fringes in an image
t increases with maximum spatial-frequency squared, these strategies (with help from more precise
m
goniometers) will be more useful as aberration-correction moves resolutions into the subangstrom
t. range.
a
m
PACSnumbers: 87.64.Bx,87.64.Ee,81.07.-b
-
d
n Contents I. INTRODUCTION
o
c
[ I. Introduction 1 Nanocrystallinematerialshaveattractedenormousin-

terestdue tophysicalandchemicalpropertiesthatdiffer
2
v II. Methods 2 fromthoseofthebulkformsthankstoproperengineering
at the atomic level1,2. Phase diagrams and crystal mor-
1
8 phologiesarefrequentlydependentonthesizeofthecrys-
III. Observations 2
2 tals in the one to ten nanometer size range3,4,5. In addi-
A. Fringe-visibility bands 2
2 tiontothissize-dependenceforthelowestenergystateof
1 B. Fringe-visibility maps 3 a structure, there is a strong tendency in the nanoparti-
2 C. Applications 3 cleregimetometastability6andnon-stoichiometry. Thus
0
/ 1. Single-particle lattice parameters 3 needs for characterizationof individualnanocrystalswill
t grow.
a 2. Random particle fringe-fingerprints 5
m A simple approach to determining the lattice (not
3. Small-tilt thickness estimates 6
atomic) structure of nanocrystals, as though they are
-
d hand specimens with visible atom columns, was pro-
n IV. Conclusions 8 posed in the late 1980’s7. It relies on the 3D reconstruc-
o tionoflatticeperiodicitiesfromimagestakenatdifferent
c Acknowledgments 8 tilts. Asindiffraction-basedtechniques8,9,10,11,12,13,14 for
:
v obtaining 3D crystallographic information, and image-
Xi References 9 based techniques in stereomicroscopy15,16,17,18,19,20,21,22
and electron tomography23,24,25,26,27,28,29,30,31, 3D infor-
r
a A. The visibility band outer half-angle 9 mation is being inferred from slices of 2D projections.
As followup, a recent direct determination of lattice pa-
rametersinthree dimensions fromimages ofa nanocrys-
B. The inner half-angle critical thickness 10
tal at two tilts32 suggested that such lattice-only analy-
sis strategies could be elegantly visualized using fringe-
C. Estimating band intersection areas 11
visibility maps that were quite sensitive to the effects of
crystal thickness.
D. The tilt-range for fringe visibility 11 In this paper we investigate visibility, versus orienta-
tion, of lattice planes which show up as fringes due to
E. Fringe-visibility rocking curves 12 scattering contrast effects when viewed nearly edge-on.
2
Suchfringecontrastoccursforexampleinelectronmicro-
scopesunderbothcoherent(phase-contrast)andincoher-
ent (z-contrast) imaging conditions. In this analysis, we
concentrate specifically on phase-contrastmechanisms.
Concepts of visibility band, and band maps, are
first introduced. Bandwidths are proportional to d-
spacing over thickness, rather than to wavelength over
d-spacing as in the case of Kikuchi33, bend-contour34,35,
electron-channeling pattern36, and backscatter-electron
diffraction37 bands in thicker specimens. Application
examples include: 1) image-based protocols for acquisi-
tionof3 linearly-independentlattice plane normalsfrom
a single randomly-oriented nanoparticle, each parallel
to an operating reflection g-vector, 2) predicting fringe
and cross-fringeabundances in a collection of randomly-
oriented particles for comparison to experimental data,
and 3) measuring local specimen thickness by observing
variations in fringe visibility during tilt.
FIG.1: Thevisibilityband(shaded)ofasetoflatticeplanes
of a spherical crystal. Such a band is so defined on a sphere
that when the electron beam direction lies in it, the lattice
II. METHODS fringes are visible. The great circle runningthrough A is the
trace of the lattice plane set. The trace runs through the
Thetransmissionelectronmicroscope(TEM)usedisa middle of a visibility band, i.e. the visibility band is sym-
Philips EM430ST with a point resolution just under 0.2 metric about the trace. The lattice fringes disappear as the
electron beam is tilted from any point along the trace in a
nm. Althoughlattice-fringeinformationisoftenavailable
directionperpendiculartothetracecirclebyananglegreater
in electron phase-constrast images out to the “damping
limit”, for simplicity we only consider fringes out to the than angle AOB=αmax, thevisibility band halfwidth.
continuous-transfer or “spherical aberration” limit for
a given image since the absence of smaller fringes may
Here d is the spacing of lattice planes, t is the crystal
be due to zeros in contrast-transfer. Hence talk here of
thickness, λ is the wavelength of the electrons, and f
“point-resolution”limits isshorthandforsayingthatde-
is a “visibility factor” on the order of 1 that empirically
ductionshereconsideronlyfringeswhosepresenceinthe
accountsforthesignal-to-noiseinthemethodusedtode-
sub-specimenelectron-wavefieldis reliablytransferredto
tect fringes. The effective radius of the reciprocallattice
the recorded image. Data from two samples were used
spot in this model is f/t. For example, when f =1 this
for experimental measurements of the probability of vi-
is the first zero in sin[πgt]/(πgt): the signal-processing
sualizing 001 zone fringes. The first contained Au/Pd
h i Fourier-transform of a unit-area boxcar function that,
nanocrystals, which were sputtered onto a thin carbon
convolvedwiththe transformofaninfinite lattice,yields
film with a Hummer IV Sputter Coater. The second
thetransformofafinitelatticeofthicknesst. Thecalcu-
sample was a nanocrystallinetungsten carbide thin film,
lation of α is based on kinematic conditions. This is
max
depositedbyplasma-enhancedchemicalvapordeposition
alsoreasonable,sinceconditionsbecomekinematicwhen
(PECVD).
the diffracted beam is about to extinguish.
For much of this paper, we assume that lattice fringes
are visible when the electron beam lies parallel to a set
III. OBSERVATIONS
of lattice planes. As shown in Appendix B, for thick
crystals under parallel illumination this may not be the
A. Fringe-visibility bands
case for α less than some α . Our assumption that
min
α =0 is typically valid for crystalsin the 10[nm]and
min
On tilting away from the edge-on view of a lattice- smaller size range.
plane, one encounters a range of incident electronangles Twosymmetriesshallbetakenintoaccountwhencon-
(e.g. relative to lattice-plane parallel) within which the sideringlatticefringevisibilityversustheincidencedirec-
lattice plane’s reciprocal lattice spots continue to inter- tion. Thesymmetriesare: anazimuthalsymmetryofthe
sectthe Ewaldsphere38. Hence lattice fringes associated electronbeamaboutthenormalofthelatticeplanes,and
with those planes remain visible. As shown in Appendix amirrorsymmetryoftheelectronbeamaboutthelattice
A, the upper bound of this range (with the largest term planes. This consideration leads to the concept of fringe
first in our “thin specimen” limit) is visibility band.
Such a visibility band is a schematic representation of
2
df λ df
α =sin−1 + 1 . (1) an ensemble of electron beam incident directions. These
max
" t 2d −(cid:18) t (cid:19) !# directions are so defined that when the electron beam is
3
In the figure, four crystal directions are marked. For
this cubic lattice, the band perpendicular to the crystal
direction of [010] is that of the (020) lattice planes, and
thatperpendicularto[001]isthebandofthe(002)lattice
planes,etc. Themapcontainsbandsof 111 , 002 and
{ } { }
220 lattice planes,with thoseofthe firsttwoclassesof
{ }
lattice plane sets drawnas shaded. The absenceof other
bandsinFig. 2suggeststhatthe smallestlatticespacing
reliably imaged by the TEM in this model is d .
220
C. Applications
As shown in Fig. 3, Kikuchi maps depend primarily
on electron wavelength while fringe-visibility maps are
affected primarily by specimen thickness. Thanks to the
reciprocal nature of diffraction, applications of Kikuchi
maps to reciprocal-space exploration are well-known.
Crystallographicinformationindirect-spaceimagesisin-
creasinglyavailable. Henceusesforfringe-visibilitymaps
FIG. 2: A fringe-visibility map for a spherical fcc crystal.
Each band width is approximately proportional to the cor- areonlybeginningtoemerge. Wediscussonlyafewhere,
responding lattice spacing and to the reciprocal lattice spot namely determination of lattice parameters, local mea-
size. Like Kikuchi maps, fringe-visibility maps reflect crys- surement of specimen thickness, and “fringe” or “cross-
tal symmetry. The latter are of special interest here as a fringe” fingerprinting of randomly-oriented particle col-
”roadmaps”toguidecrystallographicanalysesindirectspace. lections. A collection of web-interactive maps, using the
Forarandomlyorientedcrystal,thesolid-anglesubtendedby Mathematica-basedLive3DappletbyMartinKraus39,is
each band is proportional to the probability for the corre- available here40.
spondinglatticefringestobevisible,andthesolidanglesub-
tended by the cross-section of any two bands is proportional
to theprobability to get a corresponding zone-axis image.
1. Single-particle lattice parameters
along any of them, the lattice fringes are visible. As An efficient way to determine lattice parameters is
shown in Figure 1, every point on the surface of the by acquisition of three lattice vectors whose linear com-
sphere simply and elegantly represents a radially inward binations span the entire reciprocal lattice12. Two of
direction. The ensemble oforientationsfrom whicha set the three vectors may be inferred from a single zone-
of lattice planes is visible form a visibility band on the axis image. For example32 an f.c.c. WC1−x (a =
surface of the sphere. The projection of a lattice plane 4.248) nanocrystal was examined with a Gatan double-
itself is a great circle, about which the visibility band is tilt holder ( 15◦ around side-entry goniometer tilt axis,
symmetric. and 10◦ a±round the second tilt axis). The fringe-
±
visibility map (reliably seen fringes) involved only the
For thin specimens and the small λ/d of typical
shaded 111 and 002 bands in Fig. 2. Lattice pa-
electron microscopes, equation 1 simplifies to αmax ∼= { } { }
sin−1(df/t) = df/t d. Therefore visibility bands are rametersweredetermined directly from (200),(020)and
∼ ∝ (11¯1) fringes seen down [001] and [112] zone-axis direc-
different from Kikuchi bands in that the band-width is
tions.
proportional (rather than inversely-proportional) to lat-
This acquisition protocol required a total tilt range of
tice spacing. Thus we think of visibility bands as tools
35.3◦. The maximum tilt achievable with the double-
of “direct space crystallography”.
tilt holder is 35.6◦, barely higher than the tilt required.
Because of this limitation only particles at one goniome-
terextreme,withthecorrectazimuthalorientation,were
B. Fringe-visibility maps candidates for the experiment. Specifically, the [001]
zone-axis image was identified from a crystal, among
The ensemble of all the visibility bands of a spherical thousands,attheholdersettingof(15◦,9.7◦). Thiscrys-
crystal forms a fringe-visibility map. The band map not tal’s[001]zonewasazimuthallyorientedsothatthe[112]
only reveals the crystal lattice symmetry and spacing, zone axis was obtained after tilting to ( 15◦, 9.7◦).
− −
butalsoisTEM-specific,i.e. onlyresolvablelatticeplane Thusonlyasmallsubsetofparticlesusingonespecific
sets havetheir bands onthe map. Figure 2 showssucha protocol were eligible for this measurement. Improve-
map. Some examples to appreciate the crystallographic mentsinmicroscoperesolutionandgoniometerrangeare
informationinfringe-visibilitymapsaregivenasfollows. inthe works. Recentprogressinresolutionenhancement
4
FIG. 3: Comparison of Kikuchimap (left) to a fringe-visibility maps (right) for a t=4[nm] single-crystal gold sphere imaged
at100[kV].Tofirstorder,band-widthsintheformer(wheret≫tcrit)areproportionaltoλ/d,whileband-widthsinthelatter
(where t≪tcrit) are proportional to d/t . Band boldness and d-spacing are correlated.
by aberration-correction opens up myriad opportunities
for image-based nanocrystallography41,42,43. The num-
ber of protocols increases at least quadratically with the
microscope point-resolution44. Accordingly, the fringe-
visibilitybandmapwillcontainmorevisibilitybandsand
band intersections. Fig. 4 (a) and (b) show fractions of
visibilitybandmapsforan80˚AAl-crystalatapointres-
olutionof 1˚A, anda 30˚AAl-crystalat a point-resolution
of 0.6˚A, respectively. It is clear that an improvement
of continuous contrast transfer results in an increase of
the number and width of visibility bands encountered.
Thus the choice of protocols to use for lattice parameter
determination will multiply, even were the available tilt
rangetoremainfixed. Themodifiedopticsofaberration-
corrected microscopes will also allow room for a wider
range of specimen tilts.
The increase in possible analysis strategies might
therefore overwhelm one’s hope to “make sense of all
those fringes” found on tilting of a randomly-oriented
unknown crystal encountered during microscope investi-
FIG. 4: Visibility band map sections for an 80˚A Al-crystal
gation. AsshowninFig. 4(c),thisisespeciallyproblem- at a point resolution of (a) 1˚A, (b) 0.6˚A, and (c) 30˚A Al-
atic if the specimen is very thin. We therefore propose crystal at a point-resolution of 0.6˚A. Improvement in point-
aprotocolbasedonpropertiesof“afringe-visibilitymap resolution results in an increase of the number of resolvable
revealed piecemeal”, as one begins to examine fringes lattice planes, and hence that of visibility bands (from a to
seen in the candidate unknown under initially small ex- b). Reductionofcrystalsizeenablesvisualizinglatticeplanes
ploratory tilts45. over a wider range of angles, so that the width of the bands
gets larger (from b to c).
The approachis basedonone developedfor identifica-
tion of 100[nm] mineral crystallites in an unequilibrated
mixbyselectedareadiffraction46. Itmaybebrokendown
into the following steps: 2. Tilt the crystal around an axis parallel to the
fringes, until the center of the visibility band is lo-
cated.
1. Begin with a crystal orientation at which the
nanocrystalshows a setoffringes. If no fringesare
visible,thenthecrystalistobetiltedrandomlyun- 3. Tilt the crystal around an axis perpendicular to
til fringes appear. For example, imagine beginning thefringes,untilacross-fringeintersection(labeled
at the point marked “start” in the fringe visibility “ZoneA”inthemap)withanotherbandisencoun-
band map shown in Figure 5. tered.
5
TABLE I: Fringe/Zone probabilities for fcc a=4˚A; t=50˚A
g111 g200 puvw # zones
zone
h110i 2 1 .003 6
h100i 0 2 .001 3
phkl .046 .040
band
# bands 4 3
solvethesecondproblembytranslatingtokeepthenano-
crystal of interest in the field of view. The problem of
tilt hysteresis is more fundamental, and will likely have
to be addressed by independent feedback to verify stage
orientation at a given instant. Quantitative metrology
FIG.5: Visibilitybandmapsectionat0.6˚Aresolution,show-
with scanning probe microscopes requires independent
ing a trajectory for acquiring three basis fringe (reciprocal
lattice)vectorsfromarandomlyorientedunknown. Minimiz- verificationofthe probe’s lateralposition, insteadofori-
ing beam-direction errors in themeasured lattice parameters entation, but the technologies used there (e.g. optical
mandates that as wide a range of tilts as possible be used in or capacitive feedback sensing) may be relevant in both
thedetermination. cases.
4. Similarly tilt the crystal around an axis parallel 2. Random particle fringe-fingerprints
to the second band, until the center of zone A is
located. Record lattice spacings and goniometer
For a randomly-oriented nanocrystal, the probability
settings.
that a set of lattice fringes will be visible is simply pro-
5. Tiltthecrystalaroundanaxisperpendiculartothe portionaltothesolidanglesubtendedbythecorrespond-
secondband,untilasecondcross-fringeintersection ing fringe visibility band. The probability of seeing a
(labeled“ZoneB”inthe map)withthe thirdband pair of cross-fringes is proportional to the solid angle
is encountered. subtended by the corresponding visibility-band intersec-
tions. For example, for the randomly oriented spherical
6. Repeattheworkdescribedin(4)forthethirdband, f.c.c. nanocrystal whose visibility band map is shown in
topositionthebeamatthecenterofthatzoneand Figure 2, the probability of seeing (020) lattice planes
record it’s positions and lattice spacings as well. is equal to the fraction of the 4π solid angle subtended
Calculate an oriented-basis-triplet i.e. scope coor- by the (020) visibility band. For cross-fringes along the
dinates of a randomly-chosenlattice basis12. [001]zone-axis,the probabilityisequalto the fractionof
allsolidanglessubtended by the cross-sectionofthe two
7. Repeat (5)-(6) to include zones at increasingly
visibility bands centered on the [001] direction.
larger tilt angles as needed to improve accuracy of
Thus the probability for seeing a given fringe is sim-
the oriented-basis-tripletdetermination.
ply the whole-band solid angle divided by 4π, or p =
hkl
sin[α ] where α is α for hkl fringes. As shown
The basis triplet may at this point be reduced to con- hkl hkl max
{ }
in Appendix C, the solid angle subtended by a square
ventional form, assuming that measurement errors are
intersection of bands (from Eqn C3) is approximately
small enough. The precision of lattice parameter deter-
(2α )2. Hence the probabilityofseeing the associated
minationcanbeimprovedincrementallybymeasurement max
of more lattice fringes over a wider range of tilts. Best- cross-fringe pair is px ∼= 2α2max/π. More generally, for
an intersection between a band of halfwidth α and an-
fitparametersareeasilyupdatedinleast-squaresfashion 1
other of halfwidth α at an angle of φ, the cross-fringe
as a new set of fringes appear, and each fringe past the 2
probability is
first three allows one to estimate the precision in each
directionquantitatively. For example,diffractionstudies 2α α
1 2
indicatethatlatticeparameteruncertaintiesinthebeam px ∼= πsin[φ]. (2)
directionareinitially largest,andmostdependent onin-
corporation of spacings observed over a range of tilts12. Using a visibility factor of f = 0.8 estimated from
This protocol depends on one’s ability to tilt pre- Au/Pd nanoparticles evaporated on a thin carbon film,
cisely by small angles in user-chosen directions. Modern p for the [100] zone in a WC specimen was pre-
x 1−x
double-axisgoniometerscandothisusingcomputersup- dicted to be about 1/700, consistent with experimen-
port, with two caveats: (i) mechanical hysteresis leaves tal observation45. Estimation of such fringe probabili-
one with a mismatch between goniometer reading and ties in turn canbe usedto quantitativelyfingerprintcol-
the actual tilt, and (ii) the specimen will move laterally lectionsofrandomly-orientednanoparticles,accordingto
on the nanometer scale. The microscopist can generally the fringes and interspot angles in lattice images47.
6
FIG. 6: Template for fringe-visibility fingerprinting of
randomly-oriented fcc nano-particle collections. Histogram
peak heights equal p , widths equalf/t.
hkl
FIG. 7: The cross-over in abundance of cross-fringe versus
single-fringe particles, with decreasing size.
Onecanbeginthisprocesswithaprobabilitytablefor
the visible fringes expected from a given type of crys-
tal. For example, Table I lists probabilities expected for
the twolargestlattice periodicities ina collectionof50˚A viewing conditions, and thus to visibility factor f.
fcc particles with a lattice parameter or 4˚A. Here p Theprecisionoffringemeasurementsinsuchimages51
uvw
is the probability of encountering a fringe-pair for the is limited by shape anisotropy52 and other tilt
[uvw] zone. From a table like this, for the system of in- distortions53,54,55. Nonetheless the foregoing analysis
terest, one can construct a template like that in Fig. 6 works in practice, at least to first order, and provides
for plotting cross-fringe and fringe-histogram48 data. a framework for more careful study of these deviations
Figure 6 is designed to plot cross-fringe data points as well.
(two per spacing pair) and spacing histograms mea-
sured manually from multi-particle lattice fringe im-
ages. However, it is also patterned after the lay-
out of a fringe angular covariance map47. Fluctuation 3. Small-tilt thickness estimates
microscopy49, used e.g. to characterize medium-range
order in paracrystalline specimens50, historically has fo- Reduction of crystal size is well-known to result in
cusedonthe spacing-onlyor“powder”componentofthe reciprocal-space broadening of diffraction intensities as
pair-pairinformationavailableinacombinedspatialand seen in X-ray, electron and neutron diffraction. A for-
angular correlationanalysis of image data from complex mula, derived by Paul Scherrer for a thesis on col-
materials. The layout of Fig. 6, which looks at “patches loids around 1915, correlates such a broadening with
of correlated periodicity as a function of angular lag” in the inverse of the mass-weighted average grain size56.
given size range, also preserves information on the angle As shown above, decreasing size causes visibility band
between periodicities for such fluctuation analyses. Such broadening as well. Here, we discuss how band broaden-
plots may therefore prove useful in the automated anal- ing can help investigate local specimen thickness.
ysis of HRTEM images as well. Given a crystal’s size, lattice, and its orientation with
Probability expressions flowing from fringe-visibility respecttothetiltaxis,thetilt-sensitivityoflatticefringe
maps also allow one to quantify the relative abundance visibilty can be predicted. This is done by determining
ofparticleswithonlyonevisiblefringe,versusthosewith whenthereciprocallatticespotnecessarilylosesintersec-
crossfringes. Forexample,itnicelyexplainsthefactthat tion with the Ewald sphere after tilt, given the amount
2[nm] particles with cross-fringes are often more abun- oftiltusedandtheanglebetweenthefringesandthetilt
dantthan those with single fringes. This is illustratedin direction57,58. In order for the reciprocal lattice spot to
Fig. 7, which plots the fraction of particles with 110 - retain intersection with the Ewald sphere after tilt, the
h i
zone cross-fringes, versus the fraction with only a single angular deviation of the reciprocal lattice vector from
111 -fringeshowing. Our observations suggestthat the the tilt axis(or equivalentlybetweenfringe lines andthe
{ }
size at which this cross-over occurs is quite sensitive to tilt direction) must be less than an upper limit φ which
7
FIG. 8: HRTEM images of six WC1−x nanocrystals showing FIG.9: HRTEMimagesofthesixWC1−xnanocrystalsinFig
◦
lattice fringes that become invisible or remain visible after 8,afterasingletiltof14.5 aroundtheside-entrygoniometer
◦
a single tilt of 14.5 around the side-entry goniometer tilt tilt axis.
axis. The projection direction of the side-entry goniometer
tilt axis T1 is labeled at the bottom right corner. Each set
of parallel lattice-fringes in a nanocrystal has been labeled projection size of crystal A is about 48˚A. Equation 3
withbothMillerindicesandanarrowrepresentingthelattice predicts that for the given amount of tilt
fringe normal. The length of the arrow is proportional to
1/d. Here, hollow arrows are used for lattice fringes that are
predicted to become necessarily invisible, and solid arrows
φ(f =0.79,t=48˚A,d =2.453˚A)=20.6◦, (4)
for those with certain propabilities to do so, in the second 111
specimen orientation shown in Fig 9.
and
following Appendix D obeys: φ(f =0.79,t=48˚A,d002 =2.124˚A)=18.3◦, (5)
θ sinα The three lattice fringe normals deviate from the pro-
range max
sin = . (3) jection of T by 69.1◦, 0.3◦, 56.6◦, respectively. Among
2 sinφ 1
(cid:20) (cid:21) them the first and the third are larger than their corre-
Here α is given by Eqn 1. If φ is the angle between sponding limits obtained in equations 4 and 5. There-
max
an observed fringe and the tilt direction, then θ is fore,the (-1,1,-1)and(2,0,0)latticefringesarepredicted
range
the tilt-range over which that fringe is visible. Alter- tobecome invisibleaftertilt. This isshowntobe true in
natively, if θ is an experimentally-applied tilt, then Figure9. Thiswaythe invisibilityofnineoutofthirteen
range
fringes whose angle to the tilt direction is more than φ lattice fringe sets are predicted, which is consistent with
willbecomeinvisiblebecausetheirreciprocallatticespot theHRTEMobservation. TheresultsareshowninTable
will necessarily lose intersection with the Ewald sphere II.
after tilt. In this latter application, we will refer to φ in In Figure 8, the nine lattice fringe sets which are pre-
equation 3 as φ . dicted to become invisible after tilt are labeled with hol-
max
Equations 1 and 3 have been used to predict lat- low arrows. Please note that all these lattice fringe sets
tice fringe visibility after tilt for WC nanocrystals as disappear in Fig 9, which is an indication of the consis-
1−x
shown in Figure 8. In the figure, each lattice plane set tency of the model with the HRTEM observations.
has been labeled with both the Miller indices and an ar- Figure 10 shows plots of φmax versus thickness with
row representing the lattice fringe normal. The length θrange = 14.5◦ and f = 0.79 for d111 = 2.453˚A and
of the arrow is proportional to 1/d. Hollow arrows are d = 2.124˚A. Also shown in the figure are the experi-
002
usedforlattice fringes thatarepredictedto become nec- mental data from Figure 8. Hollow symbols are used to
essarily invisible, and solid arrows for those with certain label lattice fringe sets that are observed to become in-
propabilities to do so, after tilt, as shown in the Fig 9. visible after tilt as shown in Figure 9, and solid symbols
TakethosethreesetsoflatticefringesofcrystalAshown for the rest. A consistency between the model and ex-
in Fig 8, which are those of the WC (-1,1,-1), (1,1,- perimental observation is indicated, since all the hollow
1−x
1) and (2,0,0) lattice planes, as examples. The average symbols are above their corresponding curves.
8
whereφ(asabove)istheprojectedanglebetweenthelat-
TABLEII:Correlating equation3inpredictinglatticefringe
tice fringe and the tilt direction. This expressionin turn
invisibility after tilt with HRTEM observation as shown in
suggests a fractional error in thickness, per unit error in
Figures8and9. Quantitiesinthethirdandthefifthcolumns
tiltrange,of(1/t)δt/δθ tsin[φ]/(df). Forexamplethis
are obtained from Fig 8 before tilt, and column 7 contains a
≈
predictsa15percenterrorinthethickness,perdegreeof
theoretical prediction of invisibility after tilt for comparison
to theexperimental result from Fig 9 in column 8. error in tilt range, for a 50˚A thick foil with a 2˚A fringe
ID θ◦ t[˚A] (hkl) φ◦ φ◦ φ >1? Invisible? twenty degrees from the direction of tilt. This measure-
max φmax ment of specimen thickness requires only verification by
A 14.5 48 (1,1,-1) 0.3 20.6 No No
the microscope operator that tilt range over which the
(-1,1,-1) 69.1 Yes Yes fringeremainsvisibleisabout13degrees. Highprecision
(2,0,0) 56.6 18.3 Yes Yes goniometers,e.g. withverifiabletilt,andawaytoprofile
B 56 (1,1,1) 24.6 17.8 Yes Yes fringe visibility for small increments in tilt, could thus
C 42 (1,1,1) 11.8 23.4 No No make it possible to routinely measure nanocrystal thick-
D 46 (2,0,0) 57.5 19.0 Yes Yes nessatmanypointsinanimagewithonlyminoramounts
oftilt. With suchinstrumentation, experimentalprofiles
(0,2,0) 32.5 Yes Yes
offringeintensity as afunction oftilt likethose modeled
E 70 (1,1,-1) 2.2 14.5 No No
in Appendix E could open up a new world of real-time
(-1,1,-1) 65.0 Yes Yes
quantitative analysis.
(2,0,0) 58.2 13.1 Yes Yes
F 48 (1,-1,1) 67.1 20.6 Yes Yes
(1,1,-1) 43.9 Yes Yes
(2,0,0) 11.8 18.3 No No
IV. CONCLUSIONS
Fringe-visibility versus orientation for a set of lattice
planes, relative to the incident electron beam, in spec-
imens of given thickness is well represented by a “visi-
bility band”. The bands of all lattice planes reliably de-
tectedwithinthepoint-resolutionofamicroscope,during
a given exposure, make up a fringe-visibility map. Like
Kikuchi-maps in reciprocal space, fringe-visibility maps
serve as roadmaps for exploring orientation in direct
space. Weillustratehowtheycanbeusefulfordetermin-
ing the 3D lattice parameters of an arbitrary nanocrys-
tal, for fingerprinting fringe and cross-fringe abundances
inacollectionofrandomly-orientednanocrystals,andfor
determining the local thickness of crystalline specimens
with modest amounts of tilt given a sufficiently precise
FIG. 10: A plot of the maximum angular deviation of a re-
goniometer. Fringe-visibility maps, and these applica-
ciprocallatticevectorfrom thetiltaxis,asgivenbyequation
3, over crystal size. Above such a limit the reciprocal lattice tions, will be even more useful in aberration-corrected
spotnecessarilylosesintersectionwiththeEwaldsphereafter microscopescapableofsubangstromresolution,provided
tilting the crystal by 14.5◦. Experimental data from Figure attentionisgiven(duringtheirdesign)totheprecisecon-
8 and 9 are also shown. The hollow symbols, including both trol and verification of specimen orientation.
circlesandtriangles,areusedtodenotethelatticefringe sets
which are observed to become invisible after tilt, and solid
symbols for the rest lattice fringe sets. A consistency be-
tweenthemodelandtheHRTEMobservationexistssinceall
thehollow symbols are abovetheir corresponding curves.
Acknowledgments
Proofofconceptinhand,onecaneasilysolveequations
This work has benefited from discussions with L. D.
1 and 3 for specimen thickness t. Considering only the
Marks, and indirectly from support by the U.S. Depart-
dominant term, for example, one gets
mentofEnergyandtheMissouriResearchBoard,aswell
as by Monsanto and MEMC Electronic Materials Com-
θ
range
t=dfcsc[ ]csc[φ] (6) panies.
∼ 2
9
∗
[email protected] P. R.Grant (Academic Press, New York,1974).
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HALF-ANGLE
of image formation and microanalysis (Springer, Berlin,
1997), 4th ed.
36 D. C. Joy, in Quantitative Scanning Electron Microscopy, Figure11illustratesthegeometryofasimplemodelfor
edited by D. B. Holt, M. D. Muir, L. M. Boswarva, and fringe visibility in the center of an equant (e.g. spheri-
10
FIG.12: Schematicconfigurationwithanelectronbeamtrav-
eling down a zone axis, and with reciprocal lattice spots are
tangent to the Ewald sphere from the outside. The arc cen-
teredatArepresentspartoftheEwaldsphere. SegmentsBC
FIG. 11: Ewald sphere geometry for equant-particle fringe and BD represent thereciprocal lattice vectors, i.e. BC =g
visibility. HereCisthesphere’scenter,Oistheoriginofthe and BD = −g. The circles centered at C and D represent
reciprocal lattice, and A is the center of a reciprocal lattice thereciprocallatticespots. ItisobviousCDisperpendicular
spot. to AB.
to obtain
cal)particle. Visibilityrequiresthatthereciprocallattice
2
df λ df
“spot” intersect the Ewald sphere. Point O is the origin sin[α ]= + 1 . (A2)
max
of the reciprocal lattice, and two reciprocal lattice spots t 2d −(cid:18) t (cid:19) !
areshownadistance1/dawayfromthecenter. Sincethe
effective radius of reciprocal lattice spots is affected by The first term (due to beam-direction broadening of the
manyfactors,itiswrittenasf/t,wheretisthethickness reciprocal lattice spot) dominates in HRTEM of nano-
of the crystal and f is a visibility factor. This visibility crystals because d/t is typically greater than λ/d. The
factor will be a number on the order of 1 that depends equation may also be relevant to electron-channeling,
on the method of periodicity detection, the range of an- electron-backscatterdiffraction, and (with an added fac-
gles in the illuminating beam, the microscope’s response tor of 1/2) Kikuchi bands. However, in these cases the
function, and the amount of “fringe obscuring” noise in λ/dtermtypicallydominates,yieldinga1/dratherthan
the field of the image. It may need to be determined d dependence for small-angle band widths.
experimentally. As a specimen is tilted, the reciprocal
lattice spots will come in and out of contact with the
Ewald sphere. In Figure 11, the outermost edge of the
APPENDIX B: THE INNER HALF-ANGLE
reciprocal lattice spot is tangent with the Ewald sphere
CRITICAL THICKNESS
defining a critical angle, α, at which fringes of the cor-
responding spacing would be viewable in a direct-space
image. We derive α as follows. If the specimen is sufficiently thick, or electron wave-
lengths sufficiently large, images taken with a parallel
Since the radius of the Ewald sphere is 1/λ, where λ beam might also show loss of fringe visibility when the
isthe electronwavelength,wecandefine thelengthfrom specimenisorientedonthezoneaxis,i.e. betweenBragg
the center of the Ewaldsphere C to the center of the re- conditions for diffraction from either side of a set of lat-
ciprocal lattice spot A as 1/λ f/t. The distance from ticeplanes. Thisconditionwouldintroducean“invisibil-
−
the center of the Ewald sphere to the center of the re- itystripe”downthecenterofthevisibilitybanddepicted
ciprocal lattice will be 1/λ and the spacing between the in Fig. 1.
reciprocallatticespotswillbe1/d. Ifαistheangledefin-
Although the large Bragg angles and thick specimens
ing the maximum tilt of the reciprocal spot before loss
used for X-ray diffraction make this a common occur-
of fringe visibility, then we can use plane trigonometry’s
rence,itis rareinelectronlattice imagingbecauseofthe
LawofCosinesforthe complementofangleαintriangle
thinnatureofthespecimensandthesmallelectronwave-
CAO, namely
length. To confirmthis, considerthe drawingin Fig. 12.
Note that the length of segment AC is 1/λ+f/t, the
length of segment BC is 1/d, and the length of segment
2
1 f 1 1 2 π AB is 1/λ. Hence from Pythagoras’ theorem for right
= + cos[ α], (A1)
λ − t d2 λ2 − dλ 2 − triangle ABC, t becomes
(cid:18) (cid:19)