convergent beam electron diffraction

CBED was first introduced in 1939 by Kossel and Möllenstedt.{{cite journal |last1=Kossel |first1=W. |last2=Möllenstedt |first2=G. |title=Elektroneninterferenzen im konvergenten Bündel |journal=Annalen der Physik |date=1939 |volume=428 |issue=2 |pages=113–140 |doi=10.1002/andp.19394280204|bibcode=1939AnP...428..113K }} The development of the Field Emission Gun (FEG) in the 1970s,{{cite journal |last1=Crewe |first1=A. V. |last2=Isaacson |first2=M. |last3=Johnson |first3=D. |title=A Simple Scanning Electron Microscope |journal=Review of Scientific Instruments |date=February 1969 |volume=40 |issue=2 |pages=241–246 |doi=10.1063/1.1683910|bibcode=1969RScI...40..241C |url=https://digital.library.unt.edu/ark:/67531/metadc1061663/ }} the Scanning Transmission Electron Microscopy (STEM), energy filtering devices and so on, made possible smaller probe diameters and larger convergence angles, and all this made CBED more popular. In the seventies, CBED was being used for the determination of the point group and space group symmetries by Goodman and Lehmpfuh,{{cite journal |last1=Goodman |first1=P. |last2=Lehmpfuhl |first2=G. |title=Observation of the breakdown of Friedel's law in electron diffraction and symmetry determination from zero-layer interactions |journal=Acta Crystallographica Section A |date=1 May 1968 |volume=24 |issue=3 |pages=339–347 |doi=10.1107/S0567739468000677|bibcode=1968AcCrA..24..339G }} and Buxton,{{cite journal |last1=Buxton |first1=B. F. |last2=Eades |first2=J. A. |last3=Steeds |first3=John Wickham |last4=Rackham |first4=G. M. |last5=Frank |first5=Frederick Charles |title=The symmetry of electron diffraction zone axis patterns |journal=Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences |date=11 March 1976 |volume=281 |issue=1301 |pages=171–194 |doi=10.1098/rsta.1976.0024 |bibcode=1976RSPTA.281..171B |s2cid=122890943 |url=https://doi.org/10.1098/rsta.1976.0024}} and starting in 1985, CBED was used by Tanaka et al. for studying crystals structure.{{cite book |last1=Tanaka |first1=Michiyoshi |last2=Terauchi |first2=Masami |title=Convergent-Beam Electron Diffraction I |date=1985 |publisher=JEOL Ltd}}{{cite book |last1=Tanaka |first1=Michiyoshi |last2=Terauchi |first2=Masami |title=Convergent beam electron diffraction II |date=1988 |publisher=JEOL Ltd}}{{cite book |last1=Tanaka |first1=Michiyoshi |last2=Terauchi |first2=Masami |last3=Tsuda |first3=Kenji |title=Convergent beam electron diffraction III |date=1994 |publisher=JEOL Ltd}}{{cite journal |last1=Tanaka |first1=Michiyoshi |title=Convergent-beam electron diffraction |journal=Acta Crystallographica Section A |date=1994 |volume= 50|issue=3 |pages=261–286 |doi=10.1107/S0108767393010426 |bibcode=1994AcCrA..50..261T |url=https://doi.org/10.1107/S0108767393010426}}{{cite book |last1=Tanaka |first1=Michiyoshi |last2=Terauchi |first2=Masami |last3=Tsuda |first3=Kenji |last4=Saitoh |first4=Koh |title=Convergent beam electron diffraction IV |date=2002}}

By using CBED, the following information can be obtained:

• parameters of the crystal lattice,{{cite journal |last1=Jones |first1=P. M. |last2=Rakham |first2=G. M. |last3=Steeds |first3=John Wick |title=Higher order Laue zone effects in electron diffraction and their use in lattice parameter determination |journal=Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences |date=30 May 1977 |volume=354 |issue=1677 |pages=197–222 |doi=10.1098/rspa.1977.0064|bibcode=1977RSPSA.354..197J |s2cid=98158162 }} sample thickness{{cite journal |last1=Kelly |first1=P. M. |last2=Jostsons |first2=A. |last3=Blake |first3=R. G. |last4=Napier |first4=J. G. |title=The determination of foil thickness by scanning transmission electron microscopy |journal=Physica Status Solidi A |date=16 October 1975 |volume=31 |issue=2 |pages=771–780 |doi=10.1002/pssa.2210310251|bibcode=1975PSSAR..31..771K }}
• strain distribution{{cite journal |last1=Clément |first1=L. |last2=Pantel |first2=R. |last3=Kwakman |first3=L. F. Tz. |last4=Rouvière |first4=J. L. |title=Strain measurements by convergent-beam electron diffraction: The importance of stress relaxation in lamella preparations |journal=Applied Physics Letters |date=26 July 2004 |volume=85 |issue=4 |pages=651–653 |doi=10.1063/1.1774275|bibcode=2004ApPhL..85..651C }}
• defects such as stacking faults,{{cite journal |last1=Morniroli |first1=J. P. |title=CBED and LACBED characterization of crystal defects |journal=Journal of Microscopy |date=September 2006 |volume=223 |issue=3 |pages=240–245 |doi=10.1111/j.1365-2818.2006.01630.x|pmid=17059540 |s2cid=21117117 }} dislocations,{{cite journal |last1=Carpenter |first1=R. W. |last2=Spence |first2=J. C. H. |title=Three-dimensional strain-field information in convergent-beam electron diffraction patterns |journal=Acta Crystallographica Section A |date=1 January 1982 |volume=38 |issue=1 |pages=55–61 |doi=10.1107/S0567739482000102|bibcode=1982AcCrA..38...55C |doi-access= }} grain boundaries,{{cite journal |last1=Wu |first1=Lijun |last2=Zhu |first2=Yimei |last3=Tafto |first3=J. |last4=Welch |first4=D. O. |last5=Suenaga |first5=M. |title=Quantitative analysis of twist boundaries and stacking faults in Bi-based superconductors by parallel recording of dark-field images with a coherent electron source |journal=Physical Review B |date=19 September 2002 |volume=66 |issue=10 |pages=104517 |doi=10.1103/PhysRevB.66.104517|bibcode=2002PhRvB..66j4517W }} three-dimensional deformations, lattice displacements{{cite journal |last1=Wu |first1=Lijun |last2=Zhu |first2=Yimei |last3=Tafto |first3=J. |title=Picometer Accuracy in Measuring Lattice Displacements Across Planar Faults by Interferometry in Coherent Electron Diffraction |journal=Physical Review Letters |date=11 December 2000 |volume=85 |issue=24 |pages=5126–5129 |doi=10.1103/PhysRevLett.85.5126|pmid=11102202 |bibcode=2000PhRvL..85.5126W }}
• crystal symmetry information - by looking at the symmetries that appear in the CBED disks, point group{{cite journal |last1=Tanaka |first1=M. |last2=Saito |first2=R. |last3=Sekii |first3=H. |title=Point-group determination by convergent-beam electron diffraction |journal=Acta Crystallographica Section A |date=1 May 1983 |volume=39 |issue=3 |pages=357–368 |doi=10.1107/S010876738300080X|bibcode=1983AcCrA..39..357T }} and space group determination are performed.{{cite journal |last1=Goodman |first1=P. |title=A practical method of three-dimensional space-group analysis using convergent-beam electron diffraction |journal=Acta Crystallographica Section A |date=1 November 1975 |volume=31 |issue=6 |pages=804–810 |doi=10.1107/S0567739475001738|bibcode=1975AcCrA..31..804G |s2cid=98081846 |doi-access=free }}{{cite journal |last1=Tanaka |first1=M. |last2=Sekii |first2=H. |last3=Nagasawa |first3=T. |title=Space-group determination by dynamic extinction in convergent-beam electron diffraction |journal=Acta Crystallographica Section A |date=1 November 1983 |volume=39 |issue=6 |pages=825–837 |doi=10.1107/S0108767383001695|bibcode=1983AcCrA..39..825T }}
• Diagnosis of aberrations in the electron probe that limit resolution, through analysis of CBED patterns (i.e. Ronchigrams) acquired on amorphous specimens.{{cite journal | last1=Schnitzer | first1=N. | last2=Sung | first2=S.H. | last3=Hovden | first3=R.H. | title=Introduction to the Ronchigram and its Calculation with Ronchigram | journal=Microscopy Today | volume=3 | year=2019 | issue=3 | pages=12–15 | doi=10.1017/S1551929519000427 | s2cid=155224415 | doi-access=free }}

• Positions of the CBED disks are the same as the positions of the Bragg peaks and are given approximately by the relation:

$$2\; d\_\{hkl\}\; \backslash sin\backslash theta\_\{\backslash rm\; B\}\; =\; n\; \backslash lambda,$$

where $d\_\{hkl\}$ is the distance between the crystallographic planes $(h,k,l)$, $\backslash theta\_\{\backslash rm\; B\}$ is the Bragg angle, $n$ is an integer, and $\backslash lambda$ is the wavelength of the probing electrons.

• The beam convergence semi-angle $\backslash alpha$ - is controlled by the C2 aperture. The probing beam convergence semi-angle, $\backslash alpha$, is of the order of milliradians, ranging from 0.1˚ to 1˚.{{cite book |last1=Tanaka |first1=Michiyoshi |last2=Terauchi |first2=Masami |title=Convergent-Beam Electron Diffraction I |date=1985}} For small convergence semi-angle, the CBED disks do not overlap with each other, whereas for larger semi-convergence angles, the disks overlap.{{cite journal |last1=Morniroli |first1=J. P. |title=CBED and LACBED characterization of crystal defects |journal=Journal of Microscopy |date=2006 |volume=223 |issue=3 |pages=240–245 |doi=10.1111/j.1365-2818.2006.01630.x |pmid=17059540 |s2cid=21117117 |url=https://onlinelibrary.wiley.com/doi/10.1111/j.1365-2818.2006.01630.x |language=en |issn=1365-2818}}
• The diameter of a CBED disk is given by the beam convergence semi-angle $\backslash alpha$:

$$D\; =\; \backslash frac\{4\backslash pi\}\{\backslash lambda\}\; \backslash sin\; \backslash alpha.$$

• Defocus $\backslash Delta\; f$: The distance between the crossover of the probing beam and the $z$ position of the specimen is called the defocus distance $\backslash Delta\; f$. The sample can be moved along the $z$ axis. At a defocus distance, both the direct space and reciprocal space information are visible in the CBED pattern.{{cite book |last1=Cowley |first1=J. M. |title=Advances in electronics and electron physics |date=1987}}

• Conventional (C)TEM-CBED: In CTEM-CBED different shape condenser apertures are used to obtain the intensity distribution over the entire Brillouin zone.{{cite journal |last1=Tanaka |first1=Michiyoshi |title=Conventional Transmission-Electron-Microscopy Techniques in Convergent-Beam Electron Diffraction |journal=Journal of Electron Microscopy |date=1986 |doi=10.1093/oxfordjournals.jmicro.a050584}}
• Large Angle (LA)CBED: (LA)CBED is performed with a large incident angle, ranging from 1˚ to 10˚. LACBED makes it possible to obtain non-overlapping disks with a larger diameter than the one determined by the Bragg angle. With LACBED I one can obtain one selected CBED disk at a time on a detector.{{cite journal |last1=Tanaka |first1=Michiyoshi |last2=Saito |first2=Ryuichi |last3=Ueno |first3=Katsuyoshi |last4=Harada |first4=Yoshiyasu |title=Large-Angle Convergent-Beam Electron Diffraction |journal=Journal of Electron Microscopy |date=1980 |doi=10.1093/oxfordjournals.jmicro.a050262}} In LACBED II, with a slight change in the focusing conditions of the intermediate lens, bright field patterns and dark field patterns can be obtained simultaneously, without overlapping each other on the fluorescent screen.{{cite book |last1=Tanaka |first1=Michiyoshi |last2=Terauchi |first2=Masami |title=Convergent-Beam Electron Diffraction I |date=1985}} A disadvantage of LACBED is that it requires a large, flat specimen.
• 4D-STEM: In 4D-STEM a convergent probing beam is raster-scanned on a specimen in a 2D array and in each position of the array, a 2D diffraction pattern is obtained, thus generating a 4D data set. After acquisition, by using different phase techniques such as ptychography, one can recover the transmittion function and the induced phase shift.{{cite journal |last1=O'Leary |first1=C. M. |last2=Allen |first2=C. S. |last3=Huang |first3=C. |last4=Kim |first4=J. S. |last5=Liberti |first5=E. |last6=Nellist |first6=P. D.|author6-link=Peter Nellist |last7=Kirkland |first7=A. I. |title=Phase reconstruction using fast binary 4D STEM data |journal=Applied Physics Letters |date=23 March 2020 |volume=116 |issue=12 |pages=124101 |doi=10.1063/1.5143213|bibcode=2020ApPhL.116l4101O |s2cid=216342216 |url=https://ora.ox.ac.uk/objects/uuid:fda0236e-162d-4cd7-b91a-11ca5211d8c2 }} In some applications, 4D-STEM is called STEM-CBED.{{cite journal |last1=Tsuda |first1=Kenji |last2=Yasuhara |first2=Akira |last3=Tanaka |first3=Michiyoshi |title=Two-dimensional mapping of polarizations of rhombohedral nanostructures in the tetragonal phase of BaTiO 3 by the combined use of the scanning transmission electron microscopy and convergent-beam electron diffraction methods |journal=Applied Physics Letters |date=19 August 2013 |volume=103 |issue=8 |pages=082908 |doi=10.1063/1.4819221|bibcode=2013ApPhL.103h2908T}}
• Beam Rocking (BR)-CBED: With this technique, by rocking the incident beam with a rocking coil placed above the specimen, a virtual convergent beam is produced. Given that the diameter of the beam on the specimen is a few micrometers, this method has made CBED possible for materials that are susceptible to strong convergent beams. Furthermore, the large size of the illuminated specimen area and the low density current of the beam make specimen contamination insignificant.{{cite journal |last1=van Oostrum |first1=K. J. |last2=Leenhouts |first2=A. |last3=Jore |first3=A. |title=A new scanning microdiffraction technique |journal=Applied Physics Letters |date=September 1973 |volume=23 |issue=5 |pages=283–284 |doi=10.1063/1.1654890|bibcode=1973ApPhL..23..283V }}{{cite book |last1=Tanaka |first1=Michiyoshi |last2=Terauchi |first2=Masami |title=Convergent-Beam Electron Diffraction I |date=1985}}
• BR-LACBED: In this technique, in addition to the rocking coil above the specimen, there is a rocking coil placed under the projector lens, which is used to bring the preferred beam to the STEM detector. Every time the incident beam is rocked, the second coil is simultaneously driven so that the beam always falls on the STEM detector.{{cite journal |last1=Tanaka |first1=Michiyoshi |last2=Saito |first2=Ryuichi |last3=Ueno |first3=Katsuyoshi |last4=Harada |first4=Yoshiyasu |date=1 January 1980 |title=Large-Angle Convergent-Beam Electron Diffraction |url=https://academic.oup.com/jmicro/article/29/4/408/882709/LargeAngle-ConvergentBeam-Electron-Diffraction |journal=Journal of Electron Microscopy |volume=29 |issue=4 |pages=408–412 |doi=10.1093/oxfordjournals.jmicro.a050262 |issn=0022-0744}}
• Signal processing and BR-CBED: In order to enhance contrast in BR-CBED, a band-pass filter can be used that filters a certain frequency band in the CBED pattern. The combination of these two techniques makes the symmetries appearing in the patterns more clear.{{cite journal |last1=Tanaka |first1=Michiyoshi |last2=Ueno |first2=Katsuyoshi |last3=Hirata |first3=Yoshihiro |title=Signal Processing of Convergent-Beam Electron Diffraction Patterns Obtained by the Beam-Rocking Method |journal=Japanese Journal of Applied Physics |date=April 1980 |volume=19 |issue=4 |pages=L201–L204 |doi=10.1143/JJAP.19.L201|bibcode=1980JaJAP..19L.201T |s2cid=122484061 }}
• CB-LEED (Low Energy Electron Diffraction): Rocking curves are analyzed at a single energy using a convergent probe.{{cite journal |last1=Spence |first1=J.C.H. |last2=Poon |first2=H.C. |last3=Saldin |first3=D.K. |title=Convergent-Beam Low Energy Electron Diffraction (CBLEED) and the Measurement of Surface Dipole Layers |journal=Microscopy and Microanalysis |date=February 2004 |volume=10 |issue=1 |pages=128–133 |doi=10.1017/S1431927604040346|pmid=15306076 |bibcode=2004MiMic..10..128S |s2cid=46584545 }} Advantages of this method are: mapping of LEED diffraction spots into CBLEED disks, the diffraction patterns originate from a localized region of the specimen which enables the extraction of localized structural information,{{cite journal |last1=Ruben |first1=G. |last2=Jesson |first2=D.E. |last3=Paganin |first3=D.M. |last4=Smith |first4=A.E. |title=Kinematic simulation of convergent beam low-energy electron diffraction patterns |journal=Optik |date=May 2009 |volume=120 |issue=9 |pages=401–408 |doi=10.1016/j.ijleo.2007.10.006|bibcode=2009Optik.120..401R }} mapping out of the surfaces, sensitivity enhancement of small atomic displacements etc.{{cite journal |last1=Constantinou |first1=Procopios C. |last2=Jesson |first2=David E. |title=On the sensitivity of convergent beam low energy electron diffraction patterns to small atomic displacements |journal=Applied Surface Science |date=September 2019 |volume=489 |pages=504–509 |doi=10.1016/j.apsusc.2019.05.274|bibcode=2019ApSS..489..504C |s2cid=182169602 |doi-access=free }}
• Ptychography is a technique for recovering the phase of the exit electron wave. The reconstruction is done by applying an iterative phase retrieval algorithm which returns a real-space image with both phase and amplitude information. By using electron ptychography, in 2018, images of MoS₂ with an atomic resolution of 0.39 Å were reported by Jiang et al. which set the new world record for the highest resolution microscope.{{cite journal |last1=Jiang |first1=Yi |last2=Chen |first2=Zhen |last3=Han |first3=Yimo |last4=Deb |first4=Pratiti |last5=Gao |first5=Hui |last6=Xie |first6=Saien |last7=Purohit |first7=Prafull |last8=Tate |first8=Mark W. |last9=Park |first9=Jiwoong |last10=Gruner |first10=Sol M. |last11=Elser |first11=Veit |last12=Muller |first12=David A. |title=Electron ptychography of 2D materials to deep sub-ångström resolution |journal=Nature |date=July 2018 |volume=559 |issue=7714 |pages=343–349 |doi=10.1038/s41586-018-0298-5|pmid=30022131 |arxiv=1801.04630 |bibcode=2018Natur.559..343J |s2cid=119359004 }}{{cite web |title=Highest resolution microscope |url=https://www.guinnessworldrecords.com/world-records/highest-resolution-microscope |publisher=Guinness World Records}}
• Microdiffraction, nanodiffraction: In the literature, there are several terms used to refer to electron diffraction patterns that are acquired with a convergent beam. Such terms are CBED, microdiffraction, nanodiffraction etc. When the CBED technique is used for the acquisition of conventional diffraction information like lattice structure and interplanar spacing from very small areas, then the term microdiffraction is used.{{cite journal |last1=Carpenter |first1=R. W. |last2=Spence |first2=J. C. H. |title=Applications of modern microdiffraction to materials science |journal=Journal of Microscopy |date=November 1984 |volume=136 |issue=2 |pages=165–178 |doi=10.1111/j.1365-2818.1984.tb00526.x|s2cid=136906069 }} On the other hand, the term nanodiffraction is used when very small probes (< 1 nm or less in diameter) are used.{{cite book |last1=Williams |first1=David B. |title=Transmission electron microscopy: a textbook for materials science |date=2009 |publisher=Springer |location=New York |isbn=978-0-387-76501-3 |edition=2nd}}{{cite book |last1=Steeds |first1=John Wickham |last2=Hren |first2=J. J. |last3=Goldstein |first3=J. I. |last4=Joy |first4=D. C. |title=Introduction to Analytical Electron Microscopy |publisher=Plenum Press |date=1979 |pages=387}}

Since the diameter of the probing convergent beam is smaller than in the case of a parallel beam, most of the information in the CBED pattern is obtained from very small regions, which other methods cannot reach. For example, in Selected Area Electron Diffraction (SAED), where a parallel beam illumination is used, the smallest area that can be selected is 0.5 μm at 100 kV, whereas in CBED, it is possible to go to areas smaller than 100 nm.{{cite journal |last1=Champness |first1=P. E. |title=Convergent beam electron diffraction |journal=Mineralogical Magazine |date=March 1987 |volume=51 |issue=359 |pages=33–48 |doi=10.1180/minmag.1987.051.359.04|bibcode=1987MinM...51...33C |s2cid=30145465 }} Also, the amount of information that is obtained from a CBED pattern is larger than that from a SAED pattern.

Nonetheless, CBED also has its disadvantages. The focused probe may generate contamination, which can cause localized stresses. But this was more of a problem in the past, and now, with the high vacuum conditions, one should be able to probe a clean region of the specimen in minutes to hours. Another disadvantage is that the convergent beam may heat or damage the chosen region of the specimen.{{cite book |last1=Williams |first1=David B. |title=Transmission electron microscopy : a textbook for materials science |date=2009 |publisher=Springer |location=New York |isbn=978-0-387-76501-3 |edition=2nd}}

Since 1939, CBED has been mainly used to study thicker materials.

Recently, CBED was applied to study graphene{{cite journal |last1=Meyer |first1=Jannik |last2=Geim |first2= Andre K. |last3=Katsnelson |first3=M. I. |last4=Novoselov |first4=K. S. |last5=Obergfell |first5=D. |last6=Roth |first6=S. |last7=Girit |first7=C. |last8=Zettl |first8=A.|title=On the roughness of single- and bi-layer graphene membranes |journal=Solid State Communications |date= 2007 |volume=143 |issue=1–2 |pages=101–109 |doi=10.1016/j.ssc.2007.02.047|arxiv=cond-mat/0703033 |bibcode=2007SSCom.143..101M }} and other 2D monolayer crystals and van der Waals structures. For 2D crystals, the analysis of CBED patterns is simplified, because the intensity distribution in a CBED disk is directly related to the atomic arrangement in the crystal. The deformations at a nanometer resolution have been retrieved, the interlayer distance of a bilayer crystal has been reconstructed, and so on, by using CBED.{{cite journal |last1=Latychevskaia |first1=Tatiana |last2=Woods |first2=Colin Robert |last3=Wang |first3=Yi Bo |last4=Holwill |first4=Matthew |last5=Prestat |first5=Eric |last6=Haigh |first6=Sarah J. |last7=Novoselov |first7=Kostya S. |title=Convergent beam electron holography for analysis of van der Waals heterostructures |journal=Proceedings of the National Academy of Sciences |date=17 July 2018 |volume=115 |issue=29 |pages=7473–7478 |doi=10.1073/pnas.1722523115|pmid=29970422 |pmc=6055151 |arxiv=1807.01927 |bibcode=2018PNAS..115.7473L |doi-access=free }}

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