Electron lenses and aberrations: what affects the resolution in electron microscopes?
Resolution is one of the most important parameters in a scanning electron microscope (SEM). The lower the resolution, the smaller the features that can be seen. The resolution, which is typically not defined (and therefore measured) in a unique way, depends on the size of the beam when focused on the sample.
The size of the electron probe worsens when dealing with a non-ideal electron optical system, due to aberrations. What are the aberrations in electron optical systems? And how do they influence the electron probe? In this blog, we will answer these questions and give an insight into aberrations.
In a previous blog, we talked about the electron column and lenses. Typically, electron columns consist of a set of lenses that have the function of shaping the beam and focusing it on the sample surface. The size of the beam, or probe, at the sample determines the resolution of the electron microscope. But, how is the size of the probe defined? Let's find out by looking at the simplest electron optical system, shown in Figure 1.
Figure 1: The simplest electron optical system consisting of one lens at the top is the electron source, followed by a lens that focuses the beam onto the sample surface.
In this system, we know the distance between the electron source and the lens (object distance) and the distance between the lens and the sample (image distance). The image distance is also generally called the working distance and varies with the height of the sample. The ratio between the image distance and the object distance gives the magnification of the electron optical system.
where M is the magnification and ssource is the source size. So, if we consider, for example, a source with a virtual size of 50 nm and an electron optical system where the image distance is half the object distance, the contribution of the source size is 25 nm. This means that even if the system was ideal and would not suffer from aberrations, the smallest probe at the sample is 25 nm.
Aberrations originate from the fact that lenses are far from ideal. As a consequence, the probe at the sample becomes blurred and its size increases. In an electron optical system, the electron beam is affected by spherical aberrations and chromatic aberrations.
Spherical aberrations occur when the outer rays in the beam are not focused in the same plane as the inner rays. In the example of Figure 2, the outer Ray 1 is focused in a plane (Plane 1) that is closer to the lens, whereas the inner rays (Ray 3) are focused on a plane (Plane 3) that is further down.
In fact, the further from the optical axis, the more the rays are deflected, because the lens is stronger. So, if the sample is positioned in between Plane 1 and Plane 2, as shown in Figure 2, the size of the beam will be affected by the fact that all the rays are not focused on that same plane.
where k is a constant, Cs is the spherical aberration coefficient that depends on the type of lens and its geometry, and α is the half opening angle of the beam at the sample, as shown in Figure 1.
The spherical aberration contribution depends on the half opening angle of the beam to the power of 3, meaning that the closer the sample to the lens, the larger the angle, the bigger the contribution of the spherical aberration.
Figure 2: Schematics of spherical aberrations. The outer rays (Rays 3) are focused in a plane (Plane 1) that is closer to the lens.
In the electron beam, electrons don't all have the same velocity, or energy. The variation in the energy of electrons in a beam is called energy spread. Because electrons don't have the same velocity, they are not all focused in the same way.
In fact, faster electrons are more difficult to deflect, meaning that they will be focused in a plane that is more distant from the lens, as shown in Figure 3. This effect is called chromatic aberrations.
where k is a constant, CC is the chromatic aberration coefficient that depends on the strength of the lens, δU is the energy spread, V the acceleration voltage of the electron beam and α is the half opening angle of the beam at the sample, as shown in Figure 1.
The chromatic aberration contribution depends on the half opening angle of the beam, meaning that the closer the sample to the lens, the larger the angle, the bigger the contribution of the chromatic aberration.
Figure 3: Schematics of chromatic aberrations. The electrons with lower energy are focused closer to the lens than the electrons with higher energy.
What's the effect on the probe size?
Where dsource is the contribution given by the source size, ds the spherical aberrations and dc the chromatic aberrations. It is possible to calculate the total probe size for different working distances, or by placing the sample closer to the lens.
Changing the height of the sample effectively means changing the half opening angle α of the beam, depicted in Figure 1.
When the half opening angle increases, namely the sample gets further away from the last lens, the contribution of the spherical aberration dominates, whereas for the small half opening angle the main contribution is due to the demagnification of the source.
Figure 4: Total probe size and contributions from the spherical and chromatic aberrations and the demagnification of the source to the total probe size for different half opening angle (α), shown in Figure 1.
Now that you've read up on aberrations in electron optical systems like scanning electron microscopes (SEMs) and how do they influence the electron probe, exploring SEM technology in a broader sense is a logical next step. To help you do just that, we have created a comprehensible guide to SEM technology.
This guide is intended as an introductory document to electron microscopy. Its main purpose is to give an overview of why SEM technology was developed and how it works, as well as all the possibilities and applications that it provides.
Desktop SEM electron sources: why CeB6 is the right choice SEM: types of electrons, their detection and the information they provide SEM technology: the role of the electron beam voltage in electron microscopy analysis EDX analysis with a scanning electron microscope (SEM): how does it work?
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