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[[File:HrtemMg.png|thumb|High-resolution image of [[magnesium]] sample.]]
'''High-resolution transmission electron microscopy''' ('''HRTEM''') (or HREM) is an imaging mode of the [[transmission electron microscope]] (TEM) that allows for direct imaging of the atomic structure of the sample.<ref>{{cite book |title=Experimental high-resolution electron microscopy |last=Spence |first=John C. H | authorlink = John C. H. Spence |year=1988 |origyear=1980 |publisher=Oxford U. Press |location=New York |isbn=0-19-505405-9 }}</ref><ref>{{cite journal|last1=Spence|first1=J. C. H.|authorlink1 = John C. H. Spence|title=Imaging dislocation cores - the way forward|journal=Phil. Mag.|date=2006|volume=86|pages=4781–4796|doi=10.1080/14786430600776322|bibcode = 2006PMag...86.4781S |display-authors=etal}}</ref> HRTEM is a powerful tool to study properties of materials on the atomic scale, such as semiconductors, metals, nanoparticles and sp<sup>2</sup>-bonded carbon (e.g., graphene, C nanotubes). While HRTEM is often also used to refer to high resolution scanning TEM (STEM, mostly in high angle annular dark field mode), this article describes mainly the imaging of an object by recording the 2D spatial wave amplitude distribution in the image plane, in analogy to a "classic" light microscope. For disambiguation, the technique is also often referred to as phase contrast TEM. At present, the highest point resolution realised in phase contrast TEM is around {{convert|0.5|Å|nm|3|lk=on}}.<ref>{{cite journal |author1=C. Kisielowski |author2=B. Freitag |author3=M. Bischoff |author4=H. van Lin |author5=S. Lazar |author6=G. Knippels |author7=P. Tiemeijer |author8=M. van der Stam |author9=S. von Harrach |author10=M. Stekelenburg |author11=M. Haider |author12=H. Muller |author13=P. Hartel |author14=B. Kabius |author15=D. Miller |author16=I. Petrov |author17=E. Olson |author18=T. Donchev |author19=E. A. Kenik |author20=A. Lupini |author21=J. Bentley |author22=S. Pennycook |author23=A. M. Minor |author24=A. K. Schmid |author25=T. Duden |author26=V. Radmilovic |author27=Q. Ramasse |author28=R. Erni |author29=M. Watanabe |author30=E. Stach |author31=P. Denes |author32=U. Dahmen | year=2008 |title= Detection of single atoms and buried defects in three dimensions by aberration-corrected electron microscopy with 0.5 Å information limit▼
|journal= Microscopy and Microanalysis |volume=14 |pages=469&ndash;477 |doi=10.1017/S1431927608080902 |bibcode=2008MiMic..14..469K }}</ref> At these small scales, individual atoms of a crystal and [[Crystal defect|its defects]] can be resolved. For 3-dimensional crystals, it may be necessary to combine several views, taken from different angles, into a 3D map. This technique is called [[electron crystallography]].▼
 
▲'''High-resolution transmission electron microscopy''' ('''HRTEM''') (or HREM) is an imaging mode of thespecialized [[transmission electron microscope]] (TEM)s that allows for direct imaging of the atomic structure of the samplesamples.<ref>{{cite book |title=Experimental high-resolution electron microscopy |last=Spence |first=John C. H | authorlinkauthor-link = John C. H. Spence |year=1988 |origyearorig-year=1980 |publisher=Oxford U. Press |location=New York |isbn=978-0-19-505405-97 }}</ref><ref>{{cite journal|last1=Spence|first1=J. C. H.|authorlink1author-link1 = John C. H. Spence|title=Imaging dislocation cores - the way forward|journal=Phil. Mag.|date=2006|volume=86|issue=29–31|pages=4781–4796|doi=10.1080/14786430600776322|bibcode = 2006PMag...86.4781S |s2cid=135976739|display-authors=etal}}</ref> HRTEMIt is a powerful tool to study properties of materials on the atomic scale, such as semiconductors, metals, nanoparticles and sp<sup>2</sup>-bonded carbon (e.g., graphene, C nanotubes). While HRTEMthis term is often also used to refer to high resolution scanning TEMtransmission (STEMelectron microscopy, mostly in high angle annular dark field mode), this article describes mainly the imaging of an object by recording the 2Dtwo-dimensional spatial wave amplitude distribution in the image plane, in analogysimilar to a "classic" light microscope. For disambiguation, the technique is also often referred to as phase contrast TEMtransmission electron microscopy, although this term is less appropriate. At present, the highest point resolution realised in phasehigh contrastresolution TEMtransmission electron microscopy is around {{convert|0.5|Å|nm|3|lk=on}}.<ref>{{cite journal |author1=C. Kisielowski |author2=B. Freitag |author3=M. Bischoff |author4=H. van Lin |author5=S. Lazar |author6=G. Knippels |author7=P. Tiemeijer |author8=M. van der Stam |author9=S. von Harrach |author10=M. Stekelenburg |author11=M. Haider |author12=H. Muller |author13=P. Hartel |author14=B. Kabius |author15=D. Miller |author16=I. Petrov |author17=E. Olson |author18=T. Donchev |author19=E. A. Kenik |author20=A. Lupini |author21=J. Bentley |author22=S. Pennycook |author23=A. M. Minor |author24=A. K. Schmid |author25=T. Duden |author26=V. Radmilovic |author27=Q. Ramasse |author28=R. Erni |author29=M. Watanabe |author30=E. Stach |author31=P. Denes |author32=U. Dahmen | year=2008 |title= Detection of single atoms and buried defects in three dimensions by aberration-corrected electron microscopy with 0.5 Å information limit
One of the difficulties with HRTEM is that image formation relies on phase contrast. In [[phase-contrast imaging]], contrast is not necessarily intuitively interpretable, as the image is influenced by aberrations of the imaging lenses in the microscope. The largest contributions for uncorrected instruments typically come from defocus and astigmatism. The latter can be estimated from the so-called Thon ring pattern appearing in the Fourier transform modulus of an image of a thin amorphous film.▼
▲ |journal= Microscopy and Microanalysis |volume=14 |issue=5 |pages=469&ndash;477 |doi=10.1017/S1431927608080902 |pmid=18793491 |bibcode=2008MiMic..14..469K |s2cid=12689183 }}</ref> At these small scales, individual atoms of a crystal and [[Crystal defect|its defects]] can be resolved. For 3-dimensional crystals, it may beis necessary to combine several views, taken from different angles, into a 3D map. This technique is called [[electron crystallography]]tomography.
 
▲One of the difficulties with HRTEMhigh resolution transmission electron microscopy is that image formation relies on phase contrast. In [[phase-contrast imaging]], contrast is not necessarily intuitively interpretable, as the image is influenced by aberrations of the imaging lenses in the microscope. The largest contributions for uncorrected instruments typically come from defocus and astigmatism. The latter can be estimated from the so-called Thon ring pattern appearing in the Fourier transform modulus of an image of a thin amorphous film.
 
==Image contrast and interpretation==
[[File:Simulation GaN.png|thumb|Simulated HREM images for GaN[0001]]]
 
The contrast of a HRTEMhigh resolution transmission electron microscopy image arises from the [[Interference (wave propagation)|interference]] in the image plane of the [[Wave-particle duality|electron wave]] with itself. Due to our inability to record the phase of an electron wave, only the amplitude in the image plane is recorded. However, a large part of the structure information of the sample is contained in the phase of the electron wave. In order to detect it, the aberrations of the microscope (like defocus) have to be tuned in a way that converts the phase of the wave at the specimen exit plane into amplitudes in the image plane.
 
The interaction of the electron wave with the crystallographic structure of the sample is complex, but a qualitative idea of the interaction can readily be obtained. Each imaging electron interacts independently with the sample. Above the sample, the wave of an electron can be approximated as a plane wave incident on the sample surface. As it penetrates the sample, it is attracted by the positive atomic potentials of the atom cores, and channels along the atom columns of the crystallographic lattice (s-state model<ref>{{cite journal|last=Geuens|first=P|author2=van Dyck, D|title=The S-state model: a work horse for HRTEM.|journal=Ultramicroscopy|date=Dec 2002|volume=3-4|issue=3–4|pages=179–98|doi=10.1016/s0304-3991(02)00276-0|pmid=12492230}}</ref>). At the same time, the interaction between the electron wave in different atom columns leads to [[Bragg diffraction]]. The exact description of dynamical scattering of electrons in a sample not satisfying the [[weak phase object approximation]] (WPOA), which is almost all real samples, still remains the holy grail of electron microscopy. However, the physics of electron scattering and electron microscope image formation are sufficiently well known to allow accurate simulation of electron microscope images.<ref>{{cite journal | url=http://www.nature.com/nature/journal/v274/n5669/abs/274322a0.html |title=Computed crystal structure images for high resolution electron microscopy|author=O'Keefe, M. A., Buseck, P. R. and S. Iijima|volume=274|series=nature|year=1978|pages=322–324| doi= 10.1038/274322a0 | publisher=[[Nature (journal)|Nature]] | journal=Nature | issue=5669 | bibcode=1978Natur.274..322O|s2cid=4163994}}</ref>
 
As a result of the interaction with a crystalline sample, the '''electron exit wave''' right below the sample ''φ<sub>e</sub>('''x''','''u''')'' as a function of the spatial coordinate '''''x''''' is a superposition of a plane wave and a multitude of diffracted beams with different in plane [[Spatial frequency|spatial frequencies]] '''''u''''' (spatial frequencies correspond to scattering angles, or distances of rays from the optical axis in a diffraction plane). The phase change ''φ<sub>e</sub>('''x''','''u''')'' relative to the incident wave peaks at the location of the atom columns. The exit wave now passes through the imaging system of the microscope where it undergoes further phase change and interferes as the '''image wave''' in the imaging plane (mostly a digital pixel detector like a CCD camera). It is important to realize, that theThe recorded image is NOT''not'' a direct representation of the samples crystallographic structure. For instance, high intensity might or might not indicate the presence of an atom column in that precise location (see simulation). The relationship between the exit wave and the image wave is a highly nonlinear one and is a function of the aberrations of the microscope. It is described by the ''contrast transfer function''.
 
===The phase contrast transfer function===
 
The phase [[contrast transfer function]] (CTF) is a function of limiting apertures and [[Aberration in optical systems|aberrations]] in the imaging lenses of a microscope. It describes their effect on the phase of the exit wave ''φ<sub>e</sub>('''x''','''u''')'' and propagates it to the image wave. Following ''Williams and Carter'',<ref>{{cite book |title=Transmission electron microscopy: A textbook for materials science |last=Williams |first=David B. |author2=Carter, C. Barry |year=1996 |publisher=Plenum Press |location=New York |isbn=978-0-306-45324-X3 |url-access=registration |url=https://archive.org/details/transmissionelec0002will }}</ref> if we assume the WPOAweak holdsphase object approximation (thin sample), then the CTFcontrast transfer function becomes
 
: <math>CTF(u)=A(u)E(u)2\sin(\chi(u))</math>
 
where ''A('''u''')'' is the ''aperture function'', ''E('''u''')'' describes the attenuation of the wave for higher [[spatial frequency]] ''u'', also called ''envelope function''. ''χ('''u''')'' is a function of the aberrations of the electron optical system.
 
The last, sinusoidal term of the CTFcontrast transfer function will determine the sign with which components of frequency '''''u''''' will enter contrast in the final image. If one takes into account only spherical aberration to third order and defocus, χ is rotationally symmetric about the optical axis of the microscope and thus only depends on the modulus ''u'' = &#124;'''''u'''''&#124;, given by
 
: <math>\chi(u)=\frac{\pi}{2}C_s\lambda^3u^4-\pi \Delta f \lambda u^2</math>
 
where ''C<sub>s</sub>'' is the spherical aberration coefficient, ''λ'' is the electron wavelength, and Δ''f'' is the defocus. In TEMtransmission electron microscopy, defocus can easily be controlled and measured to high precision. Thus one can easily alter the shape of the CTFcontrast transfer function by defocusing the sample. Contrary to optical applications, defocusing can actually increase the precision and interpretability of the micrographs.
 
The ''aperture function'' cuts off beams scattered above a certain critical angle (given by the objective pole piece for ex), thus effectively limiting the attainable resolution. However it is the ''envelope function'' ''E('''u''')'' which usually dampens the signal of beams scattered at high angles, and imposes a maximum to the transmitted spatial frequency. This maximum determines the highest resolution attainable with a microscope and is known as the information limit. ''E('''u''')'' can be described as a product of single envelopes:
Line 42 ⟶ 44:
: <math>E_s(u) = \exp\left[-\left(\frac{\pi\alpha}{\lambda}\right)^2 \left(\frac{\delta\Chi(u)}{\delta u}\right)^2\right] = \exp\left[-\left(\frac{\pi\alpha}{\lambda}\right)^2(C_s\lambda^3u^3+\Delta f\lambda u)^2\right],</math>
 
where α is the semiangle of the pencil of rays illuminating the sample. Clearly, if the wave aberration ('here represented by ''C<sub>s</sub>'' and Δ''f'') vanished, this envelope function would be a constant one. In case of an uncorrected TEMtransmission electron microscope with fixed ''C<sub>s</sub>'', the damping due to this envelope function can be minimized by optimizing the defocus at which the image is recorded (Lichte defocus).
 
The temporal envelope function can be expressed as
Line 54 ⟶ 56:
The terms <math>\Delta I_\text{obj}/I_\text{obj}</math> and <math>\Delta V_\text{acc}/V_\text{acc}</math> represent instabilities in of the total current in the magnetic lenses and the acceleration voltage. <math>\Delta E/V_\text{acc}</math> is the energy spread of electrons emitted by the source.
 
The information limit of current state-of-the-art TEMstransmission electron microscopes is well below 1 Å. The '''[[Transmission Electron Aberration-corrected Microscope|TEAM]]''' project at Lawrence Berkeley National Laboratory resultetresulted in the first TEMtransmission electron microscope to reach an information limit of <0.5 Å in 2009 <ref>{{cite web |title=TEAM project web page |url=http://ncem.lbl.gov/team/TEAMpage/TEAMpage.html |deadurl=no |accessdateaccess-date=8 August 2013}}</ref> by the use of a highly stable mechanical and electrical environment, an ultra-bright, monochromated electron source and double-[[Sextupole magnet|hexapole]] aberration correctors.
 
===Optimum defocus in HRTEMhigh resolution transmission electron microscopy===
 
[[File:OAM CTF.png|thumb|right|CTFcontrast transfer function of the OAM{{clarify|What is OAM?|date=December 2020}} microscope]]
 
Choosing the optimum defocus is crucial to fully exploit the capabilities of an electron microscope in HRTEMhigh resolution transmission electron microscopy mode. However, there is no simple answer as to which one is the best.
 
In Gaussian focus one sets the defocus to zero, the sample is in focus. As a consequence contrast in the image plane gets its image components from the minimal area of the sample, the contrast is ''localized'' (no blurring and information overlap from other parts of the sample). The CTFcontrast nowtransfer function becomes a function that oscillates quickly with ''C<sub>s</sub>u<sup>4</sup>''. What thisThis means is that for certain diffracted beams with a given spatial frequency ''u'' the contribution to contrast in the recorded image will be reversed, thus making interpretation of the image difficult.
 
====Scherzer defocus====
Line 71 ⟶ 73:
where the factor 1.2 defines the extended Scherzer defocus. For the CM300 at [[National Center for Electron Microscopy|NCEM]], ''C''<sub>''s''</sub> = 0.6mm and an accelerating voltage of 300keV (''λ'' = 1.97 pm) ([http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/debrog2.html#c1 Wavelength calculation]) result in ''Δf<sub>Scherzer</sub> = -41.25&nbsp;nm''.
 
The point resolution of a microscope is defined as the spatial frequency ''u''<sub>res</sub> where the CTFcontrast transfer function crosses the [[abscissa]] for the first time. At Scherzer defocus this value is maximized:
 
: <math>u_\text{res}(\text{Scherzer})=0.6\lambda^{3/4} C_s^{1/4},</math>
Line 83 ⟶ 85:
 
====Lichte defocus====
To exploit all beams transmitted through the microscope up to the information limit, one relies on a complex method called '''exit wave reconstruction''' which consists in mathematically reversing the effect of the CTFcontrast transfer function to recover the original exit wave ''φ<sub>e</sub>('''x''','''u''')''. To maximize the information throughput, Hannes Lichte proposed in 1991 a defocus of a fundamentally different nature than the Scherzer defocus: because the dampening of the envelope function scales with the first derivative of ''χ(u)'', Lichte proposed a focus minimizing the modulus of d''χ''(''u'')/d''u''<ref>{{cite journal |last=Lichte |first=Hannes | year=1991 |title=Optimum focus for taking electron holograms |journal=Ultramicroscopy |volume=38 |issue=1 |pages=13&ndash;22 |doi=10.1016/0304-3991(91)90105-F }}</ref>
 
<math>\Delta f_\text{Lichte}=-0.75 C_s(u_\max\lambda)^2,</math>
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First however, both phase and amplitude of the electron wave in the image plane must be measured. As our instruments only record amplitudes, an alternative method to recover the phase has to be used. There are two methods in use today:
*'''[[Holography]]''', which was developed by [[Dennis Gabor|Gabor]] expressly for TEMtransmission electron microscopy applications, uses a prism to split the beam into a reference beam and a second one passing through the sample. Phase changes between the two are then translated in small shifts of the interference pattern, which allows recovering both phase and amplitude of the interfering wave.
* '''Through focal series method''' takes advantage of the fact that the CTFcontrast transfer function is focus dependent. A series of about 20 pictures is shot under the same imaging conditions with the exception of the focus which is incremented between each take. Together with exact knowledge of the CTFcontrast transfer function, the series allows for computation of ''φ<sub>e</sub>('''x''','''u''')'' (see figure).
 
Both methods extend the point resolution of the microscope past the information limit, which is the highest possible resolution achievable on a given machine. The ideal defocus value for this type of imaging is known as Lichte defocus and is usually several hundred nanometers negative.
 
==See also==
{{cmn|2colwidth=30em|
*[[Electron beam induced deposition]]
*[[Electron diffraction]]
*[[Electron energy loss spectroscopy]] (EELS)
*[[Electron microscope]]
*[[Energy filtered transmission electron microscopy]] (EFTEM)
*[[Scanning confocal electron microscopy]]
*[[Scanning electron microscope]] (SEM)
*[[Scanning transmission electron microscope]] (STEM)
*[[Talbot Effect]]
*[[Transmission Electron Aberration-corrected Microscope]]
*[[Transmission Electron Microscopy]] (TEM)
}}{{Library resources box▼
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== External links ==
▲{{Library resources box
|>
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{{wikibooks|Nanotechnology|Electron_microscopy#Transmission_electron_microscopy_.28TEM.29|Transmission electron microscopy (TEM)}}
* [https://web.archive.org/web/20160304001030/http://www.tem.agh.edu.pl/main/index.php?option=com_frontpage&Itemid=1&lang=en International Centre of Electron Microscopy for Materials Science]
 
==Articles==
*Topical review "Optics of high-performance electron Microscopes" [https://dx.doi.org/10.1088/0031-8949/9/1/014107 Sci. Technol. Adv. Mater. 9 (2008) 014107 (30pages)] '''''free download '''''
*[http://www.maxsidorov.com/ctfexplorer/index.htm CTF Explorer by Max V. Sidorov, freeware program to calculate the CTFcontrast transfer function]
*[http://www.tf.uni-kiel.de/matwis/amat/def_en/kap_6/backbone/r6_3_4.html High Resolution Transmission Electron Microscopy Overview]
 
==FootnotesReferences==
{{reflist|2}}
{{Electron microscopy}}
 
[[Category:Electron microscopy techniques]]
[[Category:Scientific techniques]]
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