5.2 · Electron Optical Parameters

ing voltage, so the corresponding settings would be 30 keV for high beam energy work and 1 keV for low-voltage operation. In informal conversation it is common to hear 30 kV and 30 keV used interchangeably to mean the same beam setting, and usually no confusion arises from this practice. However, in written documents such as reports of analyses or academic publications, the common error of describing the beam energy using units of kilovolts or of recording the accelerating voltage in units of kilo-electronvolts should be avoided.

Landing Energy

Aside from this possible confusion between beam energy and accelerating potential, there are other subtleties in the proper characterization of the beam energy in the SEM. Depending on the technology used by the microscope manufacturer, the electrons in the microscope may change energy more than once during their path from the electron source to the surface of the sample. Some microscopes seek to improve imaging performance by modifying the electrons’ energy during the mid-portion of the optical column. On more recent microscope models it is increasingly common to see beam deceleration options, which decrease the beam energy just before the electrons emerge from the objective lens.

Also common on modern instruments is the option to apply a voltage bias to the sample itself, thus allowing the SEM operator to increase the energy of the electrons as they approach the sample (in the case of a positive sample bias), or decrease the energy of the electrons (in the case of a negative sample bias). For example, if the electron beam emerges from the objective lens into the SEM sample chamber with a beam energy of 5 keV, but the sample has a negative voltage bias of 1 kV applied, the electrons will be decelerated to an energy of 4 keV when they impact the specimen.

The term used to describe the electron beam energy at the point of impact on the sample surface is landing energy, usually denoted by the symbol El. The physics of beam–speci- men interaction depends only on the landing energy of the electrons, not on their energy at points further up the optical path. Critically important phenomena such as the size of the excitation volume in the specimen, the number of characteristic X-ray peaks available for use in compositional measurements, or the high energy limit of continuum X-rays emitted (the Duane–Hunt limit) are all functions of the landing energy, not the initial beam energy. Because of this, it is very important for the SEM operator to understand when landing energy differs from the beam energy at the objective lens final aperture, and how to control the value of the landing energy. The details of such subtleties vary from one vendor to another, and even from one microscope model to the next, but they are invariably described in the user documentation for every instrument. Seek help from your microscope’s customer support team or an application engineer if you are not absolutely clear on how to control the landing energy on your microscope. In many situations, particularly when working with older microscopes, this distinction is not important and the terms beam energy and landing energy may be used

interchangeably without a problem, but when the distinction matters it can be crucial to accurate analysis and proper communication or reporting.

5.2.2\ Beam Diameter

Another important electron beam characteristic under the control of the SEM operator is the diameter of the electron beam, which in most cases refers to the diameter of the beam as it impacts the sample surface. Beam diameter has units of length and is frequently measured in nanometers, Ångstroms, or micrometers, depending on the size of the beam. For most SEM applications the beam diameter will fall within the broad range of 1 nm to 1 μm. It is commonly represented by

the symbol d, or a subscripted variant such as dprobe or dp. Before developing an understanding of the importance of

beam–specimen interactions, many SEM operators naively assume that the resolution of their SEM images is dictated solely by the beam diameter. While this may be true in some situations, more often the relationship between the beam diameter and the resolution is a complex one. Perhaps this explains why the exact definition of beam diameter is not always provided, even in relatively careful writing or formal contexts. The simplest model of an electron beam is one where the beam has a circular cross section at all times, and that the electrons are distributed with uniform intensity everywhere inside the beam diameter and are completely absent outside the beam diameter. In this trivial case, the beam has hard boundaries and is the same size no matter which azimuth you use to measure it. In reality the electron beam in an SEM is much more complicated. Even if you assume that the cross section is circular, electron beams exhibit a gradient of electron density from the core of the beam out to the edges, and in many cases have a tail of faint intensity that extends quite far from the central flux. It is still possible in these case to define the meaning of beam diameter in a precise way, in terms of the full width at half-­ maximum of the intensity for example, or the full width at tenth-maximum if the tails are pronounced. More careful statistical models of the beam will specify the radial intensity distribution function—a Gaussian or Lorentzian distribution, for example—and will allow for non-circularity. In most situations where such precision is not warranted, however, it will suffice to assume that the beam diameter is a single number that characterizes the width in nanometers of that portion of the beam that gives rise to the most important fraction of the contrast or sample excitation as measured at the surface of the specimen.

5.2.3\ Beam Current

Of all the electron beam parameters that matter to the SEM operator, beam current is perhaps highest on the list.

Fortunately it is a relatively simple parameter to understand since it is entirely analogous to electrical currents of the kind

found in wires, electronics, or electrical engineering. Beam current at the sample surface is a measure of the number of electrons per second that impact the specimen. It is usually measured in fractions of an ampere, such as microamperes (μA), nanoamperes (nA), or picoamperes (pA). A typical SEM beam current is about 1 nA, which corresponds to 6.25 × 109 electrons per second, or approximately one electron striking the sample every 160 ps . The usual symbol used to represent beam current is I, or i, or a subscripted variant such

as Iprobe, Ibeam, Ip, or Ib.

5

5.2.4\ Beam Current Density

Similar to the beam current, the concept of current density is relatively easy to understand and corresponds directly with the same concept in electrical engineering or electrical design. Current density in an electron beam is defined as the beam current per unit area, and it is usually represented by

the symbol J, or Jbeam. In standard units this quantity is expressed in A/m2, but there are also derived units better

suited to the SEM such as nA/nm2 or similar. The most important thing to understand about current density is that it is an areal measure, not an absolute measure; this means the current depends directly on and varies linearly with the area of the region through which the stated current density passes.

To make this concrete, let’s consider an example calculation of the current density in an electron beam. .  Figure 5.2 shows a circular beam with a diameter of 5 nm; and the total current inside the circular beam spot is 1 nA. The area of the beam is

and therefore the current density in the round beam is

a Current density = 50.9 pA/nm2

.  Figure 5.2b shows the situation if you decrease the diameter of the beam by half, from 5 nm to 2.5 nm, yet keep the same total current in the beam. Now the area has gotten smaller, yet the current is unchanged, so the current density has increased. The new current density is

Since focusing the electron beam in an SEM changes the diameter of the beam but does not change the beam current, the current density must change. As can be seen in .  Fig. 5.2, shrinking the beam width by a factor of two results in a fourfold increase in beam current density.

5.2.5\ Beam Convergence Angle, α

One of the fundamental characteristics of the electron beam found in all SEM instruments is that the shape of the beam as seen from the side is not a parallel-sided cylinder like a pencil, but rather a cone. The beam is wide where it exits the final aperture of the objective lens, and narrows steadily until (if the sample is in focus) it converges to a very small spot when it enters the specimen. A schematic of this cone is shown in

.  Fig. 5.3. The point where the beam lands on the sample is denoted S at the bottom of the cone, and the beam-defining aperture is shown in perspective as a circle at the top of the cone, with line segment AB equal to the diameter of that aperture, dapt. The vertical dashed line represents the optical axis of the SEM column, which ideally passes through the center of the final aperture, is perpendicular to the plane of that aperture, and extends down through the chamber into the sample and beyond. The sides of the cone are defined by the “edge” of the electron beam. As mentioned above in the definition of the beam diameter, this notion of a hard-edged

. Fig 5.2  a Current density in a circular electron beam; b current density if the beam diameter is reduced by a factor of two with the same current

. Fig. 5.3  a Definition of beam cone opening angle 2α; b definition of beam convergence (half) angle α

5.2 · Electron Optical Parameters

beam may not be physically realistic, but it is simple to understand and works well for our purposes here of understanding basic beam parameters.

In geometry, the opening angle of a cone is defined as the vertex angle ASB at the point of the cone, as shown in

.  Fig. 5.3a. When working with the electron optics of an SEM, by convention we use the term convergence angle to describe how quickly the electron beam narrows to its focus as it travels down the optic axis. This convergence angle is shown in .  Fig. 5.3b as α, which is half of the cone opening angle. In some cases, the beam convergence angle is referred to as the convergence half-angle to emphasize that only half of the opening angle is intended.

Generally the numerical value of the beam convergence angle in the SEM is quite small, and the electron beam cones are much sharper and narrower than the cones used for schematic purposes in .  Fig. 5.3. In fact, if you ground down and reshaped the sides of a sewing needle so that it was a true cone instead of a cylinder sharpened only at the tip, you would then have a cone whose size and shape is reasonably close to the dimensions found in the SEM.

Estimating the value of the convergence angle of an electron beam is not difficult using the triangles drawn in

.  Fig. 5.3. The length of the vertical dashed line along the optical axis is called the working distance, usually denoted by the symbol W. It is merely the distance from the bottom of the objective lens pole piece (taken here to be approximately the same plane as the final aperture) to the point at which the beam converges, which is typically also the surface of the sample if the sample is in focus. In practical SEM configurations this distance can be as small as a fraction of a millimeter or as large as tens of millimeters or a few centimeters, but in most situations W will be between 1 mm and 5 mm or so. The diameter of the wide end of the cone, line segment AB, is the aperture diameter, dapt. This can also vary widely depending on the SEM model and the choices made by the operator, but it is certainly no larger than a fraction of a millimeter and can be much smaller, on the order of micrometers. For purposes of concreteness, let’s assume W is 5 mm (i.e., 5000 μm) and the aperture is 50 μm in diameter (25 μm

in radius, denoted rapt).

From .  Fig. 5.3b we can see that triangle ASB is composed of two back-to-back right triangles. The rightmost of these has its vertex angle labeled α. The leg of that triangle adjacent to α is the working distance W, the opposite leg is

the aperture radius rapt, and the hypotenuse of the right triangle is the slant length of the beam cone, SB. From basic

trigonometry we know that the tangent of the angle is equal to the length of its opposite leg divided by its adjacent leg, or

It is no coincidence that the arc tangent of 0.005 is almost exactly equal to 0.005 radians, since a well-known approximation in trigonometry is that

Since in every practical case encountered in SEM imaging the angle will be sufficiently small to justify this approximation, we can write our estimate of the convergence angle in a much simpler form that does not require any trigonometric functions,

As mentioned earlier, this angle is quite small, approximately equal to 0.25° or about 17 arc minutes.

5.2.6\ Beam Solid Angle

In the previous section we defined the beam convergence angle in terms of 2D geometry and characterized it by a planar angle measured in the dimensionless units of radians. However, the electron beam forms a 3D cone, not a 2D triangle, so in reality it subtends a solid angle. This is a concept used in 3D geometry to describe the angular spread of a converging (or diverging) flux. The usual symbol for solid angle is Ω, and its units of measure are called steradians, abbreviated sr. Usually when solid angles are discussed in the context of the SEM they are used to describe the acceptance angle of an X-ray spectrometer, or sometimes a backscattered electron detector, but they are also important in fully describing the electron optical parameters of the primary beam in the SEM as well as the properties of electron guns.

.  Figure 5.4 shows the conical electron beam in 3D, emerging from the circular beam-defining aperture at the top

dapt

aapt

Ω

W

. Fig. 5.4  Definition of beam solid angle, Ω. The vertical dashed line represents the optical axis of the SEM, and the distance from the aperture plane to the beam impact point is the working distance, W. This is also the radius of the imaginary hemisphere used to visualize the solid angle