# Nanotechnology/Electron microscopy

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# Electron microscopy

An overview:
Electron microscopes uses electrons instead of photons, because electrons have a much shorter wavelength than photons and so allows you to observe matter with atomic resolution.
There are two general types of electron microscopes: the Scanning Electron Microscope (SEM) that scans an electron beam over the surface of an object and measures how many electrons are scattered back, and the Transmission Electron Microscope (TEM) that shoots electrons through the sample and measures how the electron beam changes because it is scattered in the sample.
Using electron beams however requires working in a vacuum environment, and this makes the instruments considerably larger and expensive. All electron microscope work under at least low pressures and usually in high vacuum chambers to avoid scattering the electrons in the gas. In environmental electron microscopes, differential pumping systems are used to actually have gasses present by the sample together with the electron beam.

## Introduction to Electron Microscopy

For imaging of nanoscale objects, optical microscopy has limited resolution since the objects are often much smaller than the wavelength of light. The achievable resolution for a wavelength $\lambda$ is often given by the diffraction limit $d$ as
    $d=0.61{\frac {\lambda }{(NA)}}$ (r.g., diffraction limit)
with numerical aperture $NA$, which can be approximated by the largest angle of incidence $\alpha$ of the wavefront towards the sample, $NA\approx \alpha$.
Since $\alpha \ll 1$ for the present purposes, we can approximate $\sin(\alpha )\approx \tan(\alpha )\approx \alpha$ and hence $NA\approx \alpha \approx r_{a}/r_{wd}$ where $r_{a}$ is the radius of the objective lens aperture and $r_{wd}$ the working distance.
Optical microscopes can often reach a resolution of about $d=200$ nm. For nanoscale resolution this is unfortunately not sufficient to distinguish for instance a single nanotube from two adhering to each other, since they have diameters of less than 100 nm.
The figure below gives an overview typical magnifications achievable by the different electron microscopes compared to a light microscope.
Electron optical systems use electrical and magnetic fields to control the electron beam. Although the law of refraction in optics is exchanged with the Lorentz force in electrodynamics, the electron optical system has similar diffraction limits as optical systems, since they depend on the wave nature of the electron beam.
One can achieve a considerable improvement in resolution with instruments such as the transmission electron microscope and the scanning electron microscope that use electrons with De Broglie wavelength much smaller than that of visible light. The De Broglie wavelength λ of an electron with momentum p is
    $\lambda ={\frac {h}{p}}={\frac {h}{\sqrt {2m_{e}E_{b}}}},$ (Eq. De Broglie wavelength)
where $h$ is Plancks constant. The electron has rest mass $m_{e}$ and energy $E_{e}=m_{e}c^{2}=511keV$.
If an electron with charge $q_{e}$ is accelerated from rest by an electrical potential $U$, to the electron beam energy $E_{b}=q_{e}U$, it will have a wavelength of 1 nm at 1 eV decreasing to 1 pm at 100 keV where it will be travelling with 50% the speed of light.
This chapter will briefly review fundamental issues for electron microscopy that are similar for SEM and TEM: the limitations imposed by the electron optical beam system in the microscope column; the interaction of the electron beam with the sample; the standard image formation method in SEM and TEM. These issues are essential to understand the results and limitations reached in SEM and TEM microscopy.
For further details, please refer to reviews of electron microscopes and their applications, such as Goldstein et al. that contains a thorough review of SEM, while Goodhew and Humphreys
is a more general introduction to both SEM and TEM.

# The Electron Optical System

For high resolution imaging, a well focused beam is required, just as in optical microscopy. Due to the short wavelength of electron beams with keV energies, as given by the #Eq de Broglie wavelength, the properties of the electron optical system and the electron emitter mainly defines the limits on the achievable beam diameter. The current density in the electron beam can be approximated by a Gaussian distribution of current density j [A/m²] as function of radius, r, from the beam center
    $j(r)=j_{0}e^{-{\frac {r^{2}}{r_{0}^{2}}}},$
with radius determined by $r_{0}$, giving a the full width half maximum $FWHM=2v(ln2)r0$. Integrating $I_{b}=\int \int j(r)rdrd\theta$ gives the total beam current
    $I_{b}=j_{0}\pi r_{0}^{2}.$
The electron optics impose a limit on the achievable beam current density and radius by the brightness of the electron emitter $\beta _{e}$, which is conserved throughout the system .
Brightness, ß, is a measure of the current per area normal to the beam direction and per element of solid angle. At the center of the Gaussian beam,
    $\beta ={\frac {j_{0}}{\pi \alpha ^{2}}}$
and the brightness is related to the current density in eq SEM Gaussian beam profile. The emitter brightness $\beta _{e}$ is determined by the type of electron emitter and the beam energy $E_{b}$
$\beta _{e}={\frac {j_{e}E_{b}}{\pi \Delta E}}$
with emission current density for W-filament sources about $j_{e}$~3 A/cm², for LaB6 sources about 100 A/cm², while field emission guns (FEG) can reach 10$^5$A/cm². The energy spread of the electrons from the sources are about ΔE~1 eV and slightly lower for FEGs. Due to conservation of the brightness in the system, the beam diameter depends on current as
$r_{0}={\frac {1}{\pi }}{\frac {r_{wd}}{r_{a}}}{\sqrt {\frac {I_{b}}{\beta _{e}}}}.$
The ideal beam probe size determined by the conservation of brightness cannot be obtained in a real system. Effects such as aberration will make the minimum achievable beam diameter larger. Equation #eq SEM beam diameter however seem to adequately describe the beam diameter for the present discussion. Apart from the additional beam widening contributions, the image detection method imposes limits on useful values for the parameters in Eq. SEM beam diameter which differ for SEM and TEM.

# Electron Range

The electron optical system sets limitations to the achievable primary beam current and radius. The expected image resolution set by the primary beam cannot be reached if the signal detected for imaging is caused by electrons scattered far in the sample. The trajectory of an electron penetrating a bulk solid is a complex trajectory due to multiple elastic and inelastic collision events. As the primary electron (PE) penetrates into the sample it will gradually change direction and loose energy in collisions. The mean free path due to elastic and inelastic collisions, $\lambda _{mfp}$, depends on the atomic number of the material and the PE energy. At 100 keV $\lambda _{mfp}=150nm$ for carbon and 5 nm for gold . For samples thinner than $\lambda _{mfp}$ the main part of the PE will pass relatively unaffected through the sample, which is the basis for TEM.
SEM can be used for thicker specimens. The electrons that escape from the sample in a new direction compared to the PE due to elastic collisions are called backscattered electrons (BSE).
For samples thicker than $\lambda _{mfp}$, the volume interacting with the scattered PE defines the range of the electrons in the material, and this is considerably larger than the minimum achievable primary beam diameters.
The electron range is about 1 µm at 10 keV for carbon, decreasing with higher atomic number for the material. Both the high energy PE and BSE generate secondary electrons (SE) by inelastic scattering events. The SE are generally defined as having energy below 50 eV while the BSE have energies up to the PE energy. The range of SE is typically 1 nm for metals and about 10 nm for insulators.
The short range of the SE make the yield of SE highly dependent on the energy lost by the PE within the SE range from the surface, and this makes high Z substances efficient generators of SE. The main emission of SE takes place in the region where the PE strikes the surface and within the SE escape depth from this region.

# Scanning electron microscopy (SEM)

* Wikipedia: Scanning electron microscopy (SEM)
In a scanning electron microscope a beam is scanned over the sample surface in a raster pattern while a signal is recorded from electron detectors for SE or BSE. The PE energy is kept relatively low (1-30 keV) to limit the interaction volume in the specimen that will contribute to the detected signal. Especially low energy PE will provide high sensitivity to surface composition as they cannot penetrate far into the sample.
The figure above showed the effect of PE penetration depth of a carbonaceous nanostructure with a gold core, where only the surface is visible at low PE energies, while the carbon becomes increasingly transparent and the core visible at high PE energies.
The low energy SE can easily be attracted and collected by a positively charged detector and are hence an efficient source for an image signal. The standard SE detector is an Everhart-Thornley (ET) detector where a positively charged grid attracts the SE and accelerates them to sufficiently high energies to create a light pulse when striking a scintillator. The light pulse is then amplified by a photomultiplier. Despite the complex construction, the ET detector is remarkably efficient, but requires large $r_{wd}$ for effective collection of the SE by the charged grid.
Another SEM detector is the in-lens detector, where SE passing through the column aperture are accelerated towards a solid state detector. The in-lens detector complements the ET by being more efficient at short $r_{wd}$.

## Environmental SEM (ESEM)

The ESEM makes it possible to use various gasses in the sample chamber of the microscope since there are narrow apertures between the sample chamber and the gun column, and a region in between that is connected to a differential pumping system. Pressures up to about 10 Torr are normally possible in the sample chamber.
The standard Everly-Thornhart SE detector would not work under such conditions since it would create a discharge in the low pressure gas. Instead a "gaseous secondary electron detector (GSD)" is used, as shown in the figure below. The GSD measures the current of a weak cascade discharge in the gas, which is seeded by the emission of electrons from the sample.
In the ESEM one can work with for instance water vapour or argon as the environmental gas, and it is possible to have liquid samples in the chamber if the sample stage is cooled sufficiently to condense water.

# Transmission electron microscopy (TEM)

* Transmission electron microscopy (TEM)
* High Resolution Transmission electron microscopy (HRTEM)
When the specimen thickness is about the mean free path, $\lambda _{mfp}$, TEM can be used to achieve high resolution images such as the image above where the atomic lattice of a gold nanocrystal is visible. Since the detected electrons are transmitted PE where the energy can be in the 100 keV range, the resolution is not limited by the issues regarding secondary electrons. The electron beam optics can be optimized for higher current densities (Eq. #eq SEM current density) at higher energies compared to SEM.
To achieve optimal imaging conditions for the thin TEM samples, the working distance has been made short. In most TEMs, the space for the sample holder is only about (5 mm)³ between the two objective lenses for the incoming and transmitted beam. Before reaching a CCD camera, the transmitted beam is sent through several magnification lenses to achieve the high magnification (500.000X is not unusual).
The image formation in TEM can be based on several principles, but practically all images used in this work were made by phase contrast imaging, here called High Resolution TEM or HRTEM. At sufficiently high brightness, electron sources can produce coherent electron beams due to the point-like emitter surface area and small energy spread. The coherent electron beam can be considered as a spherical wave propagating from the emitter and out through the electron optical system, much like a laser beam would propagate through an optical system.
The HRTEM images are often based on the interference of the electron wavefront after it has passed through the sample and reaches a CCD detector to give a phase contrast image of the sample. The image will have a resolution determined of course by the wavelength of the electrons (Eq. #eq SEM de broglie wavelength) but mainly by the imperfections of the electron optics which also perturbs the wavefront. The optimal imaging condition is for a sample thickness about $\lambda _{mfp}$, where the wavefront is only slightly perturbed by passing through the sample. TEM instruments are normally easily capable of resolving individual shells of a carbon nanotubes. The fine-tuning of the electron optical system to the required resolution can be achieved in about 30 min for many microscopes.

# Electron Holography

In special TEM microscopes, the diffracted beam can be combined with a part of the original electron beam from the electron gun, and the image that is recorded is an interference pattern that depends on how much the phase of the diffracted beam was changed. By recording such images, one can measure how the electron wave function changes as it passes through or nearby a nanostructure - and this allows you to measure the electric and magnetic fields surrounding nanostructures.

# Electron Tomography

By recording numerous TEM images of an object at many different angles, these images can in a computer be combined to create a three-dimensional model of the object. The technique is time consuming but allows you to see nanostructures in 3D.

## References

* ↑ J. Goldstein, D. Newbury, P. Echlin, D. C. Joy, A. D. Romig, C. E. Lyman, C. Fiori, and E. Lifshin. Scanning Electron Microscopy and X-Ray Microanalysis, 2nd Ed. Plenum Press, 1992.
* ↑ P. J. Goodhew and F. J. Humphreys. Electron Microscopy and Analysis, 2rd Ed. Taylor and Francis, 1988.
* ↑ S. Humphries. Charged Particle Beams. John Wiley and Sons, 1990. PDF version available at http://www.fieldp.com/cpb/cpb.html.
* ↑ P. W. Hawkes and E. Kasper. Principles Of Electron Optics. Academic Press, 1989.
* ↑ L. Reimer. Transmission electron microscopy: Physics of image formation and microanalysis, 3rd Ed. Springer-Verlag, 1993.
* ↑ P. J. Goodhew and F. J. Humphreys. Electron Microscopy and Analysis, 2rd Ed. Taylor and Francis, 1988.
* ↑ J. Goldstein, D. Newbury, P. Echlin, D. C. Joy, A. D. Romig, C. E. Lyman, C. Fiori, and E. Lifshin. Scanning Electron Microscopy and X-Ray Microanalysis, 2nd Ed. Plenum Press, 1992.
* ↑ P. W. Milonni and J. H. Eberly. Lasers. John Wiley & Sons, Inc., 1988.