(ch:scanning-electron-microscopy)=
# Scanning Electron Microscopy
(sec:SEM-TEM-and-STEM-Imaging-modes)=
## SEM, TEM, and STEM Imaging modes
In the previous chapter, we have introduced how sources and lenses for electrons work. These are the basic components of what we call the electron column: the entire system of electron optical elements that together provide a key functionality for microscopy, either illuminating the sample or projecting the illuminated sample onto a camera. What we need depends on our imaging mode, scanning or wide-field ({numref}`sec:wide-field-versus-scanning`). In a scanning system, we only need a column for illuminating the sample with a strongly demagnified image of the source (or rather, what we call the virtual source: the part of the source where all rays seem to originate from if we trace them back). The illumination is thus a finely focused spot, which we call **the probe** as we use it to probe the sample when we are scanning it over the surface. In a **Scanning Electron Microscope (SEM)**, all functionality (the electron column with source and lenses and all detectors) is located above the sample. Alternatively, we can do a wide-field illumination and then collect the electron coming from (in practise through) the sample and use an additional projection column to create an image. This is what we encounter in a **Transmission Electron Microscope (TEM)**. In a TEM the sample is thus located somewhere halfway the microscope with the illumination (or objective) column above and the projection column below.
Besides the conceptual difference in imaging mode and basic microscope layout, a major difference between SEM and TEM is the acceleration voltage of the microscope. In an SEM potential differences up to $30 \: kV$ are used and thus the energy with which electrons can arrive at the sample can range from near $0 \: eV$ up to $30 \: keV$. In a TEM, higher voltages are used and the electron energy can range from $80 \: keV$ to $300 \: keV$, with the latter used for the highest resolution microscopes.
Transmission detection can also be combined with a scanning approach. In this case, we talk about Scanning Transmission Electron Microscopy (STEM). In STEM, a transmission signal is detected that is generated by the focused probe scanning over the sample. Mostly when people refer to STEM, they refer to a TEM-like microscope operating in the high electron energy range but utilizing a focused probe illumination instead of wide-field. But alternatively, a transmission detector can be located in an SEM for STEM at lower $(≤ 30 \: keV)$ electron energies.
(sec:Basic-layout-of-a-SEM)=
## Basic layout of a SEM
The basic layout of an SEM is fairly simple (see {numref}`fig:Basic-layout-of-an-SEM`): it consists of a source unit with a first condenser lens (often called C1), followed by an aperture and a second condenser lens (C2) with another aperture, typically of variable size. The SEM column then has a final lens unit, the objective lens, which also houses the scan deflectors. The objective lens provides the focused probe on the sample. Note that depending on the mode of operation, this objective lens can use an immersion field (i.e. immersing the sample in a magnetic or electrostatic lens field) or can leave the sample field-free with the lens field restricted to the interior of the objective lens. Note also that the detectors are not indicated here, we will deal with the detectors in the chapter 5 on electron microscopy signals and detectors.
```{figure} images/Images04/Basic-layout-of-an-SEM-fixed.png
:name: fig:Basic-layout-of-an-SEM
Basic overview of electron optical elements in the column of an SEM as described in the main text.
```
The scan deflectors provide the deflection for moving the beam over the sample surface. We have previously defined the dwell time as the time that the beam spends on a single position on the sample to collect the signal for one pixel in the image (see ). In practice, the beam is actually not scanning point by point on the sample but us instead scanning over the sample in continuous fashion. The dwell time is then thus set by the integration time of the detector, i.e. the time the detector takes to collect signal that is attributed to a single pixel. Consequently, the distance traversed by the beam during this time is the pixel size. Pixel size and dwell time thus together determine the scanning speed. The dwell time will depend on the sample material, specifically the amount of contrast present and the desired signal to noise level. For biological materials this is typically on the order of one to few tens of microseconds.
(sec:Apertures-and-current)=
## Apertures and current
Apertures are present at multiple positions in the electron column. The main purpose of the apertures is to regulate beam waist and/or beam opening angle and thereby the current in the beam. For one of the aperture positions in the column, the user can typically select the aperture size in order to modify the current in the beam. Next to this, the apertures also allow the different parts of the electron column to be of different vacuum levels. As explained in {numref}`sec:electron-sources` on electron sources, the source unit needs to be on a very high vacuum level. The apertures allow the column to be differentially pumped, with a lower vacuum level $(~10^{-5} – 10^{-6} \: mbar)$ at the sample position, which makes for easier procedures for loading and mounting samples.
Due to the presence of apertures, the current in the column is not constant. Depending on where we evaluate the current, we can distinguish between source current (the current extracted from the source), the beam current (which may itself vary along the column), and the current we finally have at the sample, the probe current. For this reason, the source current is not the best parameter to quantify the emission properties of an electron source. The source brightness better indicates how much of the emitted current can be contained within a certain beam opening angle. Thus, a higher brightness source can give a higher probe current.
Of course, the probe current is most important as it tells us how many electrons arrive at the sample in a certain amount of time, which in turn determines the amount of signal generated. Current and dwell time are thus intricately linked and increasing the probe current may allow reducing the dwell time and thereby lead to lower overall measurement time. However, as we will see later, increasing the current may lead to a larger probe size and thus lower resolution.
For a Schottky FEG, the current extracted from the source can be in the order of mili-Ampère. What we finally keep as probe current for high-resolution SEM is however typically tens of pico- Ampère to a few nano-Ampère.
```{figure} images/Images04/Current-in-the-SEM-column-fixed.png
:name: fig:Current-in-the-SEM-column
Apertures in the SEM column limit the electron current. The magnitude of the electron current therefore depends on the location we consider in the column. The current in the probe will be orders of magnitude smaller than the current extracted from the source.
```
(sec:Probe-size-contributions)=
## Probe size contributions
The size of the probe that the electron column generates on the sample determines the resolution with which we can image the sample. There are four main factors that contribute to the size of the probe.
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(sec:Probe-size-versus-pixel-size-and-image-resolution)=
## Probe size versus pixel size and image resolution
Adjusting the lens and aperture settings of the SEM column allow us to optimize the beam focus and thus the probe size. As mentioned, the probe size relates to the resolution one can achieve in the measurement. However, as a user when recording an image, we set a certain image horizontal field width $(HFW)$, pixel size $(\Delta x)$, or number of pixels $(N_{pix})$, which are related via $FW = N_{pix} \cdot \Delta x $. Equivalently, in the other scan direction, we have vertical field width $VWF$ and pixel size $\Delta y$, which are ideally equal to $HWF$ and $\Delta x$ respectively so that we have isotropic pixels. The number of pixels is a power of 2, typically 1024, 2048, or 4096. Often, the pixel size is mistaken for the image resolution. First of all, we should realize that the Nyquist theorem tells us that the sampling frequency should at least be double the highest spatial frequency we still want to discern in the image. In other words, $\Delta x$ should at least be half the size of the smallest feature we want to observe. The highest resolution we can thus get in the image is by setting $\Delta x$ to half the probe size or slightly smaller. Reducing the pixel size much more will only lead to longer acquisition time with no additional information in terms of image resolution.
Another situation arises when the smallest feature in the sample, or the aimed-for resolution, is larger the probe size. In this case it is best (in terms of acquisition time) to set to half the aimed-for resolution (or slightly smaller). For instance, if we want to image a sample with $10 \: nm$ resolution, $\Delta x$ should be set to $4-5 \: nm$ even if the probe size is $3 \:n m$ or smaller. Note that the beam is scanned continuously over the sample and not point-to-point, thus each pixel is still homogeneously probed. Of course, we could in this case also work with a probe size up to $10 \: nm$ by changing the microscope column settings, with which we could have more current in the probe and thus scan with reduced dwell time.
(sec:Relation-brightness–current–probe-size)=
## Relation brightness – current – probe size
In {numref}`sec:Probe-size-contributions`, we analysed the four different contributions to the probe size in terms of the beam opening angle $\alpha$. From the discussion in the previous section, it may be clear that the probe current is also an important parameter, which does affect the probe size and thus the resolution. Using the concept of reduced brightness {numref}`subsec:brightness-and-coherence`, we can derive a relationship between the source size, or geometric, probe size contribution, the current in the probe, and α. Under conditions where diffraction and aberration contributions to the probe size can be ignored compared to the geometric contribution, this provides a direct relation for the probe current. In other conditions, we would need to assume most current is contained in the geometric contribution to the probe.
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(sec:Spatial-coherence-in-the-probe)=
## Spatial coherence in the probe
In some specific situations it may be important to have coherence ({numref}`subsec:brightness-and-coherence`) in the probe beam. In this case, a direct relationship between the current in a coherent probe and the reduced brightness can be derived. This relationship gives us the (maybe somewhat surprising) insight that spatial coherence can always be obtained if the current in the probe beam is sufficiently reduced. Of course, one should then still bear in mind that temporal coherence (the energy spread of electrons in the beam) also plays a role in the extent to which phase-sensitive imaging modes can be used.
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(sec:Coulomb-interactions)=
## Coulomb interactions
Electrons repel each other by Coulomb repulsion. The magnitude of the Coulomb force can be calculated as follows (see also {numref}`subsec:electric-charge-and-Coulomb`):
$$
F = \frac{e^2}{4 \pi \epsilon_0 r^2 }
$$ (Coulomb-force)
We see that the magnitude of the force falls off quadratically with distance between the electrons. So, the closer electrons approach each other in the beam, the stronger the force between the particles becomes. The two situations where this may happen is if we increase the current or if we lower the energy and hence the velocity of the electrons.
Coulomb repulsion between electrons can impact the performance of the electron beam in 2 ways:
- The repulsion between electrons can lead to a force component in the direction perpendicular to the beam axis, i.e. a lateral movement of the electrons. This then leads to broadening of the beam and blurring of the focus.
- The repulsion between electrons can lead to a force component in the direction parallel to the beam axis. This leads to some electrons being accelerated and others being decelerated. As a result, the energy dispersion in the beam increases, leading to more chromatic aberrations.
```{figure} images/Images04/Coulomb-repusion.png
:name: fig:Coulomb-repusion
Coulomb repulsions between electrons can lead to movement in both lateral and axial directions.
```
This is why we cannot arbitrarily increase the current in the EM probe. Even if the source brightness could still allow a higher current to be used in the probe, the equation derived in section {numref}`sec:Relation-brightness–current–probe-size` will at some point fail due to Coulomb interactions. If one sticks to operation conditions in the user interface of the microscope supplier, with current used in operation typically ranging between tens of $pA$ to a few $nA$, Coulomb interactions can be safely ignored.
(sec:Electron-beam-opening-angles)=
## Electron beam opening angles
In section {numref}`sec:Probe-size-contributions` and from the above consideration of Coulomb interactions, we have seen that beam opening angles in electron microscopy are very small compared to light microscopy. Typical opening angles in EM are in the order of $mrad$’s. From light microscopy, we may know that a very large opening angle, leads to a very small focal depth. In EM, the focal depth can be considerably larger compared to high-resolution (high NA) light microscopy. This may be clear when we look at typical SEM images like images of small bugs: the entire bug appears in focus, despite the macroscopic size of the bug. This is due to the small opening angle of the beam, leading to a relatively large focal depth.
```{figure} images/Images04/bug.jpg
---
name: fig:bug
width: 500px
height: 300px
align: center
---
From *Art and Architecture of Insects* by D. M. Phillips (2014). Copyright 2014 by [ForeEdge]. No further use allowed.
```
(sec:Astigmatism)=
## Astigmatism
EM lens fields may be sensitive to imperfections, i.e. deviations from ideal, symmetric behaviour. Because of the small beam opening angles and thus relatively small beam waist in a macroscopic column, the electron beam may easily go a bit off axis. This may also lead to asymmetricuuu interactions on the beam. These imperfections lead to aberrations and one of the prominent aberrations present in an electron microscope is astigmatism. In the case of astigmatism, lens operation is stronger in one of the lateral directions compared to the other. If we take a plane perpendicular to the axis along which electrons travel (i.e. a plane parallel to the sample plane), we can indicate these two lateral directions as $x$ and $y$ direction. The direction along the axis is then the z direction. An astigmatic beam may experience a stronger lens field in for example the $x$ direction compared to the $y$ direction. This means the focus in the $x$ direction will be located earlier on the optical axis compared to the $y$ direction focus. Further along the optical axis, the y direction will be in focus and the $x$ direction will be out of focus. The apparent, symmetric focus will occur at a location in between the positions where $x$ and $y$ are in focus. However, this spot will be blurred in both directions (see {numref}`fig:Astigmatism`).
```{figure} images/Images04/Astigmatism.png
:name: fig:Astigmatism
Astigmatism depicted schematically with representations of the different x- and y-focus positions along the optical axis & microscopic images obtained at the x-focus (left), the apparent best focus (center) and the y-focus (right).
```
Whether there is astigmatism in a beam can be spotted if the sample is moved through focus (moving from the x-focus position to the y-focus position): this leads to elongated defocusing in the beam going from one direction to the other direction. All SEMs have an option for adjusting astigmatism in this way in order to find the best focus conditions and doing so is standard practice in aligning the SEM.