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Lecture 11: TEM: Beam - sample
interaction
Contents
1 Introduction
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2 TEM Resolution and wavelength
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3 Sample thickness in TEM
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4 Electron interaction with matter
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5 Scattering angles in TEM
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6 Mechanism of elastic scattering
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6.1 Atomic scattering factor . . . . . . . . . . . . . . . . . . . . . 13
7 Inelastic scattering
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7.1 Secondary electrons . . . . . . . . . . . . . . . . . . . . . . . . 17
8 Beam damage in TEM
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Introduction
Electron microscopy is an important characterization technique that uses
electron beams to provide information on the sample. There are 2 main
variants
1. Transmission electron microscopy (TEM) - here the electron beams
are transmitted through the specimen and the image is formed on the
opposite side of the electron source
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2. Scanning electron microscopy (SEM) - we make use of secondary electrons (obtained after interaction of sample with the primary beam) to
form the image.
The most important reason for developing TEM is the limited spatial resolution of the light microscope. Typical light microscopes have spatial resolution
of the order of the wavelength of light (few hundred nm). This is not good
enough to resolve individual lattice planes or even atoms. For this we need
wavelength of the order of ˚
A.
One option would be x-rays. X-ray used for diffraction have wavelength
around a few ˚
A (Cu Kα wavelength is 1.54 ˚
A). So x-rays can be used to form
images with atomic resolution. The problem with using x-rays is that it is
not possible to focus the beams using lenses. There are no lenses available,
especially for hard x-rays. There is some x-ray microscopy with soft x-rays
(wavelength tens of nm) mainly used for biological samples. X-rays are also
used for tomography (3D measurement). But there is no x-ray imaging available for hard materials. Hence we need to turn to electron beams for imaging
with atomic resolution.
A brief history of the TEM. Louis de Broglie (in 1925) was the first to theorize that electrons had wave like characteristics. This meant that they could
be diffracted, similar to electromagnetic radiation. In 1927, Davisson and
Germer showed that electron diffraction is possible which lead to the concept of electron microscopy. In 1932, Knoll and Ruska proposed the idea
and built the first electron lens. Ruska won the Nobel prize for this in 1986
along with Binning and Rohrer, who invented the STM in late 1970s. A picture of the earliest TEM is shown in figure 1. Contrast that with a modern
TEM (Titan from FEI) in figure 2. The first commercial TEMs were built
mainly in the 1950s (after WWII). There are a number of TEM companies
like Philips, JEOL, Hitachi, RCA, and FEI.
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TEM Resolution and wavelength
Resolution is defined as the smallest distance between two points/lines that
can be distinguished. For the naked eye, the resolution is around 0.1-0.2 mm.
Resolution also defines the highest useful magnification of an instrument.
Anything more that that is just empty magnification. The Rayleigh criteria
defines resolution (δ) as
0.61λ
(1)
δ =
µ sin β
where λ is the wavelength of the radiation, µ is the refractive index of the
medium, and β is the semi-angle of collection of the magnifying lesn. The
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Figure 1: Earliest TEMs built in 1930s by Knoll and Ruska. Taken from
Transmission Electron microscopy - Williams and Carter.
term µ sin β is called the numerical aperture of the lens. If the numerical
aperture is approximately 1 then the resolution is 0.61λ. For visible light
with λ of 400 nm, this translates to around 240 nm. This number is still 3
orders of magnitude higher than the typical lattice spacing in metals. The
solution is to use electrons with wavelength comparable to lattice spacing.
The relation between λ and energy (E) for electromagnetic radiation is
λ =
hc
E
(2)
where h is Planck’s constant and c the velocity of light. This cannot be
used for electrons since they have a finite mass and cannot reach the speed
of light. Consider an electron beam accelerating through some potential V .
The energy of the electron is given by eV , where e is the charge of an electron
(1.6 × 10−19 C). This is also the origin of the energy units eV (which is the
energy gained by an electron after accelerating through 1 V and is equal to
1.6 × 10−19 J). This energy can be related to the kinetic energy and hence
the velocity by the equation
r
2eV
v =
(3)
me
where me is the mass of the electron (9.1 × 10−31 kg). Using the de Broglie
relation, the wavelength is related to the momentum and hence the velocity
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Figure 2: FEI Titan TEM. Taken from Transmission Electron microscopy Williams and Carter.
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Figure 3: Electron beam interaction for a thin sample. Taken from Transmission Electron microscopy - Williams and Carter.
by
h
me v
λ =
(4)
˚, which
Thus, an electron accelerated by 100 kV (105 V) has a λ of 0.04 A
is less than the interatomic spacing. Thus using the Rayleigh criteria, in
equation 1, this should give a sub-˚
A resolution! This is rarely the case, the
limiting factor being the quality of the electron lens. Currently, sub nm
resolution can be achieved while in STEM mode sub-˚
A is possible at the
highest quality!
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Sample thickness in TEM
Electrons are strongly interacting with matter and produce a wide range of
secondary signals. The summary of electron interaction with a thin transmitting sample is shown in figure 3. A number of secondary signals are produced,
some of which are useful for related electron microscopy techniques but for
TEM the direct beam and the elastically scattered electrons are used. Because of the strong electron-matter interaction the sample has to be thin for
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Figure 4: Electron escape depth as function of energy.
Taken
from http://engineering.siu.edu/frictioncenter/cafs-courses/surface-contactmechanics/lecture-5.php.
the beam to be transmitted. How thin? In x-ray diffraction the beam can
penetrate a depth of around few µm. In electron microscopy the sample needs
to be less than 100 nm thick, with typical thicknesses of 30-50 nm, for the
beam to transmit through the sample. The escape depth of electrons from
different metals as a function of energy is plotted in figure 4. This requires
extensive sample preparation. Another related issue with thin samples and
high energy electrons is the beam damage is possible, especially for biological
samples. Also, in TEM we are imaging the entire 3D sample (30-50 nm) but
projecting it onto a 2D screen. Thus, the image obtained is a 2D projection
of electron interaction with a 3D sample. So image interpretation, especially
for high resolution, is not straightforward.
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Electron interaction with matter
Consider the interaction between the electron beam and the solid, as shown
in figure 3. The diagram is true for all electron microscopy no just TEM. In
the case of SEM, the sample is thick so that there is no transmitted beam.
This is shown in figure 5. Not all the signals shown in figures 3 and 5 can be
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Figure 5: Electron beam interaction for a bulk sample. Taken from Transmission Electron microscopy - Williams and Carter.
used at the same time. Also, both figures show only electron signals. There
are other signals that are also generated. In the case of TEM, the complete
list of signals is shown in figure 6. The characteristic x-rays, shown in figure 6
can be used for chemical analysis. This technique is called energy dispersive
x-ray analysis (EDAX) and is similar to the technique on x-ray fluorescence.
In some TEM designs, due to constraints of space, the EDAX detector is
usually held off-axis to the main electron beam column and hence cannot be
used when the sample is being imaged.
The interaction between the electron beam and the sample is coulombic.
Since electrons are negatively charged the incoming beam can interact strongly
with the electron cloud in the solid and also the positively charged nucleus.
In contrast x-rays are EM radiation and they only interact with the electron
cloud. In TEM, for imaging purposes, only the forward scattered electrons
are of interest. There are two main types of scattered radiation
1. Elastic - this represents coherent scattering (mainly) with no loss of
energy. There is also a phase relation with the incident radiation.
2. Inelastic - the energy of the scattered electrons is lower than the incident beam. These are also incoherent radiation with no phase relation
with the incident radiation.
An important approximation in the TEM, especially for thin samples, is the
single scattering event. The idea is that the incident electron beam undergoes only one scattering event as it passes through the sample. This is also
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Figure 6: Complete electron beam interaction for a thin sample.
used for image simulation in TEM using the kinematic theory. For samples
around 20-30 nm thick kinematic scattering is a valid assumption but fails
for thicker samples. Then, dynamic scattering theory is used for simulations
of TEM images and diffraction patterns.
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Scattering angles in TEM
In the TEM Bragg scattering angles are very small. This is due to the small
wavelength of the electron beam. Consider Bragg’s law
2dsinθ = λ
(5)
Rearranging this, the Bragg angle is related to the ration of the beam wavelength to the lattice constant. For a 100 keV electron beam λ = 0.039 ˚
A.
˚
Typical d-spacings for metals is around 2 A. So using equation 5 sinθ is
0.010. The corresponding Bragg angle is 0.572◦ or 0.091 rad. So for a TEM,
equation 5 can be modified to read
sinθ ≈ θ =
8
λ
2d
(6)
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Figure 7: Aperture of different sizes in a TEM.Taken from Transmission
Electron microscopy - Williams and Carter.
Figure 8: Fraunhofer scattering from a single slit of width w.Taken from
Transmission Electron microscopy - Williams and Carter.
Thus, the scattered beams are very close to the central beam. Contrast
this with XRD where 2θ are generally around 20 - 150◦ . This is due to the
difference in the wavelengths.
Since we are interested in beams close to the central axis of the microscope
(corresponding to the direct beam) we can make use of apertures to limit the
electron beam so that oblique radiation is blocked. Aperture is a metal piece
with a circular hole of a specific diameter. A number of different apertures
are used, as shown in figure 7. Problem with an aperture is that there will be
diffraction at the edges when an electron beam passes through. This is called
Fraunhofer diffraction. Fraunhofer diffraction produces a broad intensity
pattern. This is shown in figure 8, for electron scattering from a single slit
of width w. For a TEM, the electron wavelength is much smaller than w but
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Figure 9: Airy ring pattern due to diffraction from a slit of width w.Taken
from Transmission Electron microscopy - Williams and Carter.
still there will be a finite intensity width. For a circular disk of diameter w,
the peak width is given by
λ
(7)
θw = 1.22
w
This is called an Airy disk pattern and the observed Airy rings are shown
in figure 9. Airy patterns are named after Sir George Biddell Airy an English mathematician from 1800’s. For a smaller aperture (D↓) the width,
from equation 7,(θw ) ↑. The resolution suffers since scattering from a point
source is not spread over a finite width due to the Fraunhofer scattering.
But a smaller aperture also improves the depth of the filed. Thus, there is
a tradeoff between the two parameters. Another way to improve the resolution, a smaller λ is preferable, since once again, using 7, the θw is smaller.
But electron with higher energy can also cause beam damage, especially for
biological samples.
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Mechanism of elastic scattering
In TEM, we are concerned mainly with elastic scattering since this is the
signal we want to measure. Elastic scattering is where the energy of the
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electron remains unchanged. Usually elastic scattering is coherent, but not
always. We can look at scattering from
1. An individual atom
2. A group of atoms i.e. solid (crystalline).
This is similar to the treatment used for x-ray interaction with a solid.
Consider electron scattering with an isolated atom. There is a Coulombic
interaction with the electron cloud or with the nucleus. Electron nucleus interaction is very strong and results in high angle scattering. Electron-electron
interaction is weaker and results in low scattering angles. Also, e− -e− scattering need not be always elastic. There could also be inelastic scattering.
Scattering of a electron with an isolated atom is summarized in figure 10.
Define a scattering cross-section for the e− -nucleus and the e− -e− interaction. It is defined as the hypothetical area around a scatter which describes
the probability of radiation or other particles (in TEM it is electrons) being
scattered. Dividing the scattering cross section by the total area gives the
probability of scattering. The scattering cross section for e− -e− interaction
is given by
e
(8)
σe− −e− = πre2 = π( )2
Vθ
where V is the voltage through which the incoming electron beam is accelerated (defines its energy and velocity), re is the radius of the electron cloud,
and θ is the scattering angle. Equation 8 can be modified for an electronnucleus interaction to give
σe− −n = πrn2 = π(
Ze 2
)
Vθ
(9)
where rn is the radius of the nucleus and Z is the atomic number.
A more detailed explanation for the scattering by a nucleus can be given by
considering the Rutherford scattering cross section. The angular dependence
of scattering, i.e. σ(θ) as a function of the solid angle of scattering Ω is given
by
dσ(θ)
e2 Z 4
=
(10)
dΩ
16E02 sin4 2θ
This can be integrated over Ω to give the total elastic scattering from a
sample (due to contribution from the nucleus). Integrating equation 10 gives
σnucleus = 1.62 × 10−24
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Z2
θ
cot2
2
E0
2
(11)
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Figure 10: Electron beam interaction with an isolated atom showing both
high and low angle scattering from the nucleus and the electron cloud. Taken
from Transmission Electron microscopy - Williams and Carter.
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This gives the scattering cross section for a given incident angle θ. If we
consider the total elastic scattering from a sample (integrating θ over 0 to π)
we can get the total elastic scattering from a sample of thickness t. This is
given by
NA (ρt)σnucleus
(12)
σt =
A
where NA is Avogadro’s number, A is the area of the sample. The quantity ρt
is called the mass thickness of the specimen. In scattering from the nucleus
the mass thickness is proportional to the square of the atomic number (Z).
So higher the value of Z, more is the scattering. This is the source of one of
the contrast mechanisms in the TEM. This also explain why gold (Z=79) is
one of the best elements for recording in the TEM. Au is also inert so that
oxide formation is not an issue. If we take into account screening from the
inner shell elements, can replace Z by Zef f .
6.1
Atomic scattering factor
If we consider scattering from the nucleus and the electron cloud we can define
an atomic scattering factor, f (θ). This depends on the electron wavelength
(λ), atomic number (Z), and the scattering angle (θ). The atomic scattering
factor term is similar to what was defined earlier for X-ray diffraction. It has
the following characteristics
1. Scattering is maximum for small θ. As θ increases, f drops.
2. As atomic number increases, f is higher. For θ equal to zero, f is
usually very close to the atomic number (Z).
3. As wavelength increases, f decreases.
Figure 11 shows the variation in f for Cu, Al, and Au as a function of sinλ θ .
Au has a higher atomic number so its atomic scattering factor is noticeably
higher. In all 3 materials f decreases as θ increases or λ increases. For a
group of atoms in the form of a crystal (periodic arrangement) we have a
structure factor, F , that depends on the location and type of atoms. This
is again similar to X-ray diffraction and we will look at it during diffraction
contrast.
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Inelastic scattering
In inelastic scattering, the incoming electron loses energy to the sample. The
electron beam that is transmitted then has a lower energy (higher wavelength) and is also incoherent with respect to the incoming beam. There are
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Figure 11: Atomic scattering factor for 3 elements as a function of θ and λ.
Taken from Transmission Electron microscopy - Williams and Carter.
different mechanisms by which energy is lost and Electron Energy Loss Spectroscopy (EELS) makes use of the energy loss to provide chemical and bonding information about the sample. Electron energy loss is sometimes accompanied by formation of x-rays. These can be the continuous (Bremsstrahlung
radiation) or Characteristic x-rays. The formation of characteristic x-rays is
shown schematically in figure 12. The mechanism is similar to that in X-ray
diffraction, except that the core hole is created by electrons. In figure 12
the electrons that have lost energy can be used in EELS. The cross-section
for x-ray production depends on the over voltage, i.e. the ratio of electron
energy (E) to the electron energy needed for core hole ionization (E0 ). In
TEM, E is usually around 100-200 keV while E0 is usually less than 10 keV .
The over voltage is then around 10-20. Figure 13 is a plot of the ionization cross section vs. the over voltage. Ideally, we would want to be in the
regime where the cross section does not depend on the over voltage so that
any quantitative analysis is simplified. The complementary process to x-ray
production is Auger electron formation and is summarized in figure 14. This
dominates especially for light elements. Auger process will be discussed later.
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Figure 12: Characteristic x-ray production in TEM. Taken from Transmission Electron microscopy - Williams and Carter.
Figure 13: Ionization cross section vs. the over voltage. Taken from Transmission Electron microscopy - Williams and Carter.
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Figure 14: Auger electron emission in TEM. Taken from Transmission Electron microscopy - Williams and Carter.
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7.1
Secondary electrons
Secondary electrons (SE) are inelastic electrons that are ejected by the primary beam. There are 2 main types of secondary electrons
1. Slow SE - electrons ejected from the valence and/or conduction band
have low energy (¡50 eV). They are called slow SE.
2. Fast SE - electron ejected from the inner shell require a significant
energy transfer. These electrons are called Fast SE.
Secondary electrons are used mainly in SEM (Scanning electron microscopy).
They provide topographical information (slow SE) and also Z information.
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Beam damage in TEM
Electron beams in the TEM can cause damage to the specimen. Electron
energy loss from the beam can be transferred to the sample. This can be
used to increase the vibration of the lattice atoms i.e. generate heat. This is
shown in figure 15. Apart from heating the electron beam can also damage
chemical bonds especially in polymeric and biological bonds where the bonds
are weak. This is one of the reasons, why biological samples are usually
imaged at cryogenic temperatures with low dose of electrons. Electron beam
can also cause sputtering of the sample by direct atom displacement.
The most common form of beam damage is heating. Heating is severe since
the sample thickness is very small, of the order of tens of nm, so that the
thermal mass is small. The problem is particularly severe for non-conducting
samples, where rise in temperature can be hundreds of ◦ C.
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Figure 15: Lattice oscillations in a TEM due to the electron beam. Taken
from Transmission Electron microscopy - Williams and Carter.
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