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[[File:HrtemMg.png|thumb|High-resolution image of [[magnesium]] sample.]]
'''High-resolution transmission electron microscopy''' ('''HRTEM''') 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. |year=1988 |origyear=1980 |publisher=Oxford U. Press |location=New York |isbn=0-19-505405-9 }}</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 |author= C. Kisielowski, B. Freitag, M. Bischoff, H. van Lin, S. Lazar, G. Knippels, P. Tiemeijer, M. van der Stam, S. von Harrach, M. Stekelenburg, M. Haider, H. Muller, P. Hartel, B. Kabius, D. Miller, I. Petrov, E. Olson, T. Donchev, E. A. Kenik, A. Lupini, J. Bentley, S. Pennycook, A. M. Minor, A. K. Schmid, T. Duden, V. Radmilovic, Q. Ramasse, R. Erni, M. Watanabe, E. Stach, P. Denes, 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=469–477 |doi=10.1017/S1431927608080902 }}</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'''
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=469–477 |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|
▲One of the difficulties with
==Image contrast and interpretation==
[[File:Simulation GaN.png|thumb|Simulated HREM images for GaN[0001]]]
The contrast of a
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|
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).
===The phase contrast transfer function===
The phase [[
: <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
: <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
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:
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: <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
The temporal envelope function can be expressed as
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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
===Optimum defocus in
[[File:OAM CTF.png|thumb|right|
Choosing the optimum defocus is crucial to fully exploit the capabilities of an electron microscope in
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
====Scherzer defocus====
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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 nm''.
The point resolution of a microscope is defined as the spatial frequency ''u''<sub>res</sub> where the
: <math>u_\text{res}(\text{Scherzer})=0.6\lambda^{3/4} C_s^{1/4},</math>
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====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
<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
* '''Through focal series method''' takes advantage of the fact that the
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.▼
▲* '''Through focal series method''' takes advantage of the fact that the CTF 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 CTF the series allows for computation of ''φ<sub>e</sub>('''x''','''u''')'' (see figure).
▲Both methods extend the point resolution of the microscope 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|colwidth=30em|
*[[Electron beam induced deposition]]
*[[Electron diffraction]]
*[[Electron energy loss spectroscopy]] (EELS)
*[[Electron microscope]]
*[[Energy filtered transmission electron microscopy]]
*[[Scanning confocal electron microscopy]]
*[[Scanning electron microscope]]
*[[Scanning transmission electron microscope]]
*[[Talbot Effect]]
*[[Transmission Electron Microscopy]] (TEM)▼
*[[Transmission Electron Aberration-corrected Microscope]]
}}{{Library resources box▼
▲{{Library resources box
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{{wikibooks|Nanotechnology|Electron_microscopy#Transmission_electron_microscopy_.28TEM.29|Transmission electron microscopy
==Articles==
*Topical review "Optics of high-performance electron Microscopes" [
*[http://www.maxsidorov.com/ctfexplorer/index.htm CTF Explorer by Max V. Sidorov, freeware program to calculate the
*[http://www.tf.uni-kiel.de/matwis/amat/def_en/kap_6/backbone/r6_3_4.html High Resolution Transmission Electron Microscopy Overview]
==
{{reflist|2}}
{{Electron microscopy}}
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[[Category:Scientific techniques]]
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