# Nanotechnology/Electron microscopy#The Electron Optical System

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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.

! A very simple sketch of a Transmission Electron Microscope (TEM) and
Scanning Electron Microscope (SEM) compared to an optical transmission
microscope and a cathode ray tube (CRT) TV screen - both systems have
many things on common with the electron microscope. The optical
microscope uses lenses to control the lights pathway through the system
and is in many ways built up like a TEM - only the TEM uses
electromagnetic lenses to direct the beam of electrons. The CRT uses
electromagnetic lenses as the TEM and SEM to control the electron beam,
and generates an image for the viewer by scanning the beam over a
fluorescent screen - in the same way the a SEM generates an image by
scanning the electron beam over a small
sample. and Scanning Electron Microscope (SEM) compared to an optical transmission microscope and a cathode ray tube (CRT) TV screen - both systems have many things on common with the electron microscope. The optical microscope uses lenses to control the lights pathway through the system and is in many ways built up like a TEM - only the TEM uses electromagnetic lenses to direct the beam of electrons. The CRT uses electromagnetic lenses as the TEM and SEM to control the electron beam, and generates an image for the viewer by scanning the beam over a fluorescent screen - in the same way the a SEM generates an image by scanning the electron beam over a small sample."){width="300"}

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.

!The different methods for microscopy cover a range of magnification
roughly indicated by the bars in the figure. The resolution of optical
microscopy is limited to about 200 nm. a) SEM image of the head of an
ant facing a microfabricated chip with a pair of microfabricated
grippers. The grippers are barely visibly at the tip of the arrow. b)
SEM image of a gripper approaching a large bundle of carbon nanotubes.
c) Closeup in SEM of the gripper and nanotubes. d) TEM image of a carbon
nanotube suspended between two grippers. e) TEM closeup of the shells of
carbon atoms in a carbon nanotube. On the nanometer scale this
particular carbon nanotube does not show a well defined carbon shell
structure. SEM image of the head of an ant facing a microfabricated chip with a pair of microfabricated grippers. The grippers are barely visibly at the tip of the arrow. b) SEM image of a gripper approaching a large bundle of carbon nanotubes. c) Closeup in SEM of the gripper and nanotubes. d) TEM image of a carbon nanotube suspended between two grippers. e) TEM closeup of the shells of carbon atoms in a carbon nanotube. On the nanometer scale this particular carbon nanotube does not show a well defined carbon shell structure."){width="400"}

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 $E_{e}=m_{e}c^2=511 keV$.

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. [^1] that contains a
thorough review of SEM, while Goodhew and Humphreys [^2] 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 [^3].

Brightness, ß, is a measure of the current per area normal to the beam
direction and per element of solid angle [^4]. At the center of the
Gaussian beam,

\<div id=\"eq SEM current density\>

```{=html}
</div>
```
```{=html}
<div id="eq beam brightness">
```
```{=html}
</div>
```
$$\beta=\frac{j_0}{\pi \alpha^2}$$

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 $E_{b}$ [^5]

$\beta_{e}=\frac{j_{e}E_{b}}{\pi \Delta E}$

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

```{=html}
<div id="eq SEM beam diameter">
```
```{=html}
</div>
```
$r_{0}=\frac{1}{\pi}\frac{r_{wd}}{r_{a}}\sqrt{\frac{I_{b}}{\beta_{e}}}.$

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 $\lambda_{mfp}=150 nm$ for
carbon and 5 nm for gold [^6]. 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.

!Overview of electron electron scattering processes in bulk and
tip-shaped specimens. The PE are scattered within the interaction
volume, defined the electron range in the material. The range is longer
than the mean free path $\lambda_{mfp}$. The SE have a very short range,
and only those created within that range from the surface can escape the
material. This defines the SE escape
depth.{width="300"}

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 [^7].

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.

!The electron range increases with beam energy. The internal structure
of the EEBD deposits can be examined at high electron beam energies in
SEM. At 5 kV with shallow penetration depth, the surface of the tips is
clearly visible while at higher energies a core of more dense material
becomes increasingly visible. At 100 keV and above, TEM images can
achieve atomic resolution where the lattice planes in nanocrystals such
as the gold nanocrystal in (c). The gold crystal is embedded in
amorphous carbon with no clear lattice
pattern.. The gold crystal is embedded in amorphous carbon with no clear lattice pattern."){width="300"}

# 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)

!Simple sketch of an Environmental Scanning Electron Microscope (ESEM),
where a differential pumping system with two pressure limiting apertures
between the ultra high vacuum SEM column and the low vacuum sample
chamber allows high pressures up to 10 hPa around the sample. This is
enough to have liquid water at moderate cooling of 5 deg.
C., where a differential pumping system with two pressure limiting apertures between the ultra high vacuum SEM column and the low vacuum sample chamber allows high pressures up to 10 hPa around the sample. This is enough to have liquid water at moderate cooling of 5 deg. C."){width="300"}

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.

! Two examples of images from an ESEM. Taken with a Philips XL-30 FEG.
The first shows a electron beam deposited nanowire between two
microelectrodes that has burnt after sustaining a high bias current. The
other shows a multiwall carbon nanotube sample. Shorter working
distances often improves image quality and so does a low beam current
but it also increases the image acquisition
time{width="300"}

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)

! A Philips EM 430
TEM{width="100"}

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 [^8]. 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.

!TEM images of the same nanostructure using standard \'bright field\'
TEM vs HAADF STEM. The sample is a gold nanoparticle containing
environmental electron beam deposited rod.
{width="300"}

# 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


[^1]: 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.

[^2]: P. J. Goodhew and F. J. Humphreys. Electron Microscopy and
    Analysis, 2rd Ed. Taylor and Francis, 1988.

[^3]: S. Humphries. Charged Particle Beams. John Wiley and Sons, 1990.
    PDF version available at <http://www.fieldp.com/cpb/cpb.html>.

[^4]: P. W. Hawkes and E. Kasper. Principles Of Electron Optics.
    Academic Press, 1989.

[^5]: L. Reimer. Transmission electron microscopy: Physics of image
    formation and microanalysis, 3rd Ed. Springer-Verlag, 1993.

[^6]: P. J. Goodhew and F. J. Humphreys. Electron Microscopy and
    Analysis, 2rd Ed. Taylor and Francis, 1988.

[^7]: 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.

[^8]: P. W. Milonni and J. H. Eberly. Lasers. John Wiley & Sons, Inc.,
    1988.
