The handle http://hdl.handle.net/1887/63484 holds various files of this Leiden University dissertation.
Author: Geelen, D.
Title: eV-TEM: transmission electron microscopy with few-eV electrons Issue Date: 2018-05-31
Chapter 4
Spectroscopy I: Elastic processes
S
low electrons have a longer wavelength than fast electrons, as the De Broglie wavelength decreases when the electron kinetic energy is increased (λ = h/√2mE). In low-energy electron microscopy the wavelength can be of the order of the atomic spacing or even, in the case of (multilayer) graphene, comparable to the sample thickness. This leads to quantum interference effects that strongly influence the electron transmissivity and reflectivity. Since the wavelength of an electron depends on energy, occurrence of interference depends strongly on the electron energy. LEEM is therefore the perfect instrument to study these effects, as it allows for accurate energy control over three orders of magnitude in the range of 0 to 100 eV with 0.1 eV steps. In this chapter we present reflection and transmission experiments with multilayer graphene. Our results show that the behaviour of the slowest electrons is dominated by elastic effects and that inelastic effects take over at higher energies. This chapter will focus on the elastic interactions.Inelastic interactions are the subject of chapter 5.
There are several reasons to study (multilayer) graphene. First, graphene is a material that shows fascinating electrical and mechanical properties. Second, graphene is atomically thin, therefore a high electron transmission is expected.
(Bio)Organic materials, such as DNA or proteins, can be deposited on graphene to enable eV-TEM with minimal substrate interference. Thus, it is important to have a good understanding of the graphene substrate itself.
71
In the previous chapter, we presented eV-TEM and LEEM micrographs of multilayer graphene. These show strong contrast in the transmitted and reflected LEE intensity between regions. In this chapter, we study the energy dependence of this contrast. In 2008, Hibino et al. [1] showed that the layer thickness of graphene grown on SiC(0001) can be determined from oscillations in the energy dependent reflectance spectra. Later, Feenstra et al. found that the minima in the reflectance spectra are a consequence of high-transmission states [2] (Feenstra et el. refer to these states as high-transmission resonances).
Until now such states have only been studied in reflection measurements.∗ With eV-TEM we can directly observe such states in transmission. At the same energy as the minima in the reflection spectrum, we expect to see maxima in the transmission, just like in the toy model presented in chapter 1.
The toy models do not take the three-dimensional (quasi-two-dimensional) nature of graphene into account. We expect this to be important. First, the two-dimensional periodic crystal lattice of graphene gives rise to diffracted beams influencing the reflected and transmitted signals. Second, in the in- plane direction, a 2D dispersion relation can be expected. This will influence the transmission and reflection probability of non-normal incidence LEE. In section 4.4 we present a method that allows us to probe the dispersion of the interlayer states that are part of the unoccupied band structure.
4.1 Low-energy electron reflection spectra
In figure 4.1, two LEEM-reflection micrographs of freestanding graphene are shown at 2.3 eV and 4.0 eV. The graphene in this sample was grown on copper with chemical vapor deposition and transferred to a PtPd-coated Si3N4
window. The preparation of such a sample is discussed in detail in section 2.3.1.
In figure 4.1a, areas with different intensities can be identified, labeled A, B and C. Comparing the two micrographs we find that the different regions do not exhibit the same energy dependence. Region C goes from bright to dark, while region B does the opposite and region A does not exhibit such intensity variations. We can determine the reflectivity spectrum of each region, pixel by pixel, by acquiring a sequence of micrographs at different energies.
∗Müllerová et al. have observed a local transmission maximum below 10 eV but could not distinguish the individual resonances [3, 4].
4.1. Low-energy electron reflection spectra 73
2.3 eV
500 nm C
B A
(a)LEEM 2.3 eV
4.0 eV
500 nm C
B A
(b)LEEM 4.0 eV
Figure 4.1:LEEM-reflection micrographs of freestanding graphene measured in bright field at different energies: (a) is obtained with 2.3 eV and (b) with 4.0 eV. The graphene is suspended over a 2 µm hole in a PtPd-coated Si3N4membrane. The preparation of these samples is discussed in chapter 3. Regions A, B, and C correspond respectively to one, two and three graphene layers.
0 2 4 6 8 10 12
Energy (eV)
0.0 0.2 0.4 0.6 0.8 1.0
Reflectivity
A: One layer B: Two layers C: Three layers
2.3 eV
500 nm
Figure 4.2:Normalized reflection measured as a function of energy from different areas on freestanding graphene. The black, blue and green curves are obtained on monolayer, bilayer and trilayer graphene, respectively.
The spectra, obtained from energy-filtered micrographs of the indicated re- gions, are plotted in figure 4.2. Several regimes can be identified. First the intensity is high (mirror mode) and rapidly decreases as the electrons cross 0eV energy and reach the sample. Between ∼ 0 and 7 eV, in regions B and C, the signal oscillates, while in region A the intensity steadily drops. Region C shows two minima, while region B shows only one.
Hibino et al. [1] demonstrated that the number of minima is determined by the number of stacked graphene layers, and that the reflectivity spectrum of a stack of n graphene layers has n − 1 minima [2, 5–7]. Therefore we conclude that regions A, B and C corresponds to single, double, and triple layer graphene, respectively.
The minima in the spectra are a consequence of high-transmission states similar to those found in the toy model (see chapter 1). These have never been observed directly in transmission. With eV-TEM we expect to find maxima in the transmission spectra.
4.2 Transmission states measured with eV-TEM
The energy dependent electron transmittance can be investigated with eV- TEM. An electron source is placed behind the sample and the electron landing energy is determined by the potential difference between the cathode and the sample (see chapter 2). Just like in reflection, a series of micrographs at different energies can be made to obtain transmission spectra.
Figure 4.4b shows transmission spectra of regions with one to four graphene layers (this is obtained with graphene on a molybdenum coated Si3N4window).
These spectra are normalized by determining the transmission through an open hole without graphene close to this region using the method described in appendix E. In the transmission measurement there is no intensity at E < 0eV. This is in mirror-mode where the electrons travel back into the eV- TEM illumination optics. For positive electron energies, i.e. above the mirror- mode transition, electrons are transmitted. The eV-TEM electron source has a broader energy width than the source used for the reflection experiments, which causes a broadening of all features in the eV-TEM spectrum.
4.2. Transmission states measured with eV-TEM 75
5.1 eV
500 nm A
B C
B A
(a)eV-TEM 5.1 eV
500 nm 10.9 eV
C
B A
(b)eV-TEM 10.9 eV
Figure 4.3:eV-TEM transmission micrograph of freestanding (multilayer) graphene.
This is the same region as the reflection micrographs presented in figure 4.1. Regions A, B and C correspond to areas with one, two and three graphene layers. (a) is obtained with an electron energy of 5.1 eV and (b) with 10.9 eV electrons.
The transmission spectra of multilayer graphene show oscillations. The spec- trum obtained from the double-layer region shows a maximum in intensity at an electron energy that coincides with the minimum in the reflection spectrum in figure 4.4a. This confirms our expectations and shows that the observed oscillations are indeed related to high-transmission states. The spectrum of triple layer also has a clear maximum that coincides with the first minimum in the reflection spectrum. However, at the energy at which we expect to find a second maximum, we find a shoulder. The quadruple layer transmission spectrum also has shoulders that coincide with the minima in the reflectance spectrum.
Now that we are able to measure the reflection and the transmission spectra we can learn more about the interactions of LEE with a material by comparing the two. Interestingly, the oscillations in the transmission spectra are qualitatively similar to the resonances in the toy model with loss processes. Here too, the oscillations in reflection are more pronounced than in transmission where they are suppressed. They are therefore manifested as a shoulder-like feature in the transmission spectrum. This implies that electrons undergo loss processes as they interact with the graphene layers. This is the subject of the next chapter.
In (multilayer) graphene, the loss is due to (inelastic) electron scattering. In
0 2 4 6 8 10 12
Energy (eV)
0.0 0.2 0.4 0.6 0.8 1.0
Reflectivity
One layer Two layers Three layers Four layers
4.0 eV
500 nm
2.4 eV
500 nm
(a)Reflection
0 2 4 6 8 10 12
Energy (eV)
0.0 0.2 0.4 0.6 0.8 1.0
Transmissivity
One layer Two layers Three layers Four layers
2.4 eV
500 nm A
A
B C
B A
B CC A
B C
16 eV
500 nm
(b)Transmission
Figure 4.4: (a)Reflection and (b) transmission spectra from multilayer graphene (on a molybdenum coated Si3N4window). The black, blue, green and red curves were obtained on 1, 2, 3, and 4-layer graphene, respectively. The color-matched dots in the real-space images indicate where the spectra have been taken.
4.3. Higher energy states 77
the energy-filtered imaging configuration (see section 2.1.4) any electron that transferred enough energy or momentum to fall outside the energy slit or contrast aperture, will not be detected. This has the consequence that the transmission spectra, T (E), are not the exact complement of the reflection spectra R (E), i.e. T +R 6= 1. This becomes more apparent at higher electrons energies as we will see in the next section.
4.3 Higher energy states
So far, we have only studied resonances at energies < 10 eV. With increasing energy both the reflected and transmitted signal intensities drop precipitously.
To record high-quality spectra over a broad energy range we use the High Dynamic Range (HDR) method explained in appendix C.
In figure 4.5 HDR transmission and reflection spectra are presented. The spec- tra are now plotted on a logarithmic intensity scale to show the low intensity at high energies, enabling us to discern the minima around 20, 30, and 50eV.
However, for the higher-energy minima, we do not observe additional splitting, in contrast to what we find for the lower-energy resonances in multilayer graphene. The separate resonances are broadened and thus harder to discrim- inate compared to the lower energy resonances. This is caused by inelastic losses, as can also be seen from the model in chapter 1. This will be further discussed in chapter 5. Figure 4.5a clearly shows an overall exponential decay of the reflected signal. The transmission spectra also decrease with energy.
Interestingly, this only happens until ∼ 30 eV, after which the transmission increases. This is in accordance with the expectation due to the universal curve, presented in chapter 1.
As expected, the reflectance spectrum of freestanding monolayer graphene does not have minima like the other spectra. However, it does show a drop of intensity around 30 eV. In addition to the resonances, the spectra in figure 4.5b all have a minimum at around 28 eV. This coincides with the intensity drop in the monolayer spectrum. Because the features appear in both mono- and multilayer graphene, this cannot be a consequence of interlayer resonances.
0 10 20 30 40 50 60 70
Energy (eV)
10−4 10−3 10−2 10−1 100
Reflectivity
One layer Two layers Three layers
2.3 eV
500 nm
(a)LEEM
0 10 20 30 40 50 60 70
Energy (eV)
10−4 10−3 10−2 10−1 100
Transmissivity(Arb)
One layer Two layers Three layers
0 10 20 30 40 50 60 70
Energy (eV)
10−4 10−3 10−2 10−1 100
Transmissivity
One layer Two layers Three layers
(b)eV-TEM
Figure 4.5: (a)Low-energy electron reflection and (b) transmission spectra for multi- layer graphene measured with HDR (explained in appendix C).
4.4. Unoccupied band structure; ARRES 79
This reduction in both the reflected and transmitted signal is caused by the appearance of higher-order diffracted beams. In the bright-field configura- tion the spectra are only obtained from the specular (0, 0)-spot, of which the intensity reduces when higher-order beams form. The drop in intensity happens at an energy of ∼ 28 eV. However, the higher-order diffraction spots only appear in the diffraction pattern at around 33 eV. This discrepancy is a consequence of the work function of the material, i.e. the energy difference φbetween the Fermi and the vacuum level. The work function of graphene is about 4.5 eV and differs between mono, double and triple-layer graphene by about 0.1 eV [8]. An electron entering the material gains an energy φ, while one leaving the material has to overcome the surface potential barrier and therefore loses the same amount of energy. Thus, when a 28 eV electron enters the material, it gains an energy φ such that diffracted beams can form inside the material. However, the out-of-plane momentum component, ~k⊥, is not sufficient to overcome the surface potential barrier. The diffracted beam is therefore confined in the in-plane direction (analogous to total internal reflection).
4.4 Unoccupied band structure; ARRES
∗The measured reflection and transmission spectra are closely related to the unoccupied band structure above the vacuum level. The band structure of a material consists of all allowed electronic states. The states below the vacuum level are bound and states above the vacuum level are unbound. eV-TEM can be used to directly probe the unoccupied states above the vacuum level. The spectra shown so far are all obtained with electron illumination under normal incidence, i.e. with zero in-plane momentum. That means that we probe the unoccupied band structure at the Γ-point (the center of the Brillouin zone).
The high-transmission states yield clear features in the reflectance spectra. We have seen that the maxima in the transmission spectra correspond to minima in reflection†. Since in reflection the in-plane momentum of the incident electrons can easily be controlled with the deflectors in the illumination sys- tem we can also obtain reflection spectra with non-zero in-plane momenta,
∗The findings in this section have been published in [9].
†n.b. not all minima in the transmission spectrum correspond with maxima in the reflection spectrum. Some of the features are not related to a high-transmission state, as discussed in the previous section.
i.e. away from the Γ-point. As a result, we can probe the unoccupied band structure of multilayer graphene, grown on SiC(0001). We call this technique Angle-Resolved Reflection Electron Spectroscopy (ARRES). This allows us to measure the dispersion of the interlayer states in multilayer graphene.
Figure 4.6a shows a normal incidence reflection micrograph of graphene grown on the SiC(0001) surface. Again, a sequence of micrographs with different electron energies is made so that reflectance spectra can be obtained pixel by pixel. Figure 4.6b shows such spectra on double, triple and quadruple layer graphene∗. The in-plane momentum of the incident electron beam can be monitored in the diffraction plane. Figure 4.6e shows the diffraction pattern of graphene on SiC with normal incidence illumination. Here the (0, 0)-spot (the specular spot) is in the center of the Brillouin zone, referred to as the Γ- point. The other spots in this figure are Moiré spots resulting from a mismatch between the graphene layer and the SiC substrate [10] (these therefore do not occur on freestanding graphene). To obtain a reflectance spectrum with non-normal incident electrons, the specular spot is moved in the diffraction plane. In figure 4.6f the spot is moved in the direction of the M-point and in figure 4.6g the specular spot is placed on the K-point.
For each of the ARRES measurements of this SiC(0001) sample, shown in figures 4.7a, b, and c, sequences of micrographs have been made for 21 different values of kk. Every column in these figures is a reflectance spectrum, the intensity is represented in a color-scale. These data have been obtained from the regions with different layer count indicated in figure 4.6a.
In figure 4.7a the ARRES measurement of double layer graphene is shown. At the Γ-point, where kk = 0, the high-transmission state (i.e. low reflection) can clearly be identified (indicated by a white dot). The ARRES measurements on three and four layer graphene, figures 4.7b and 4.7c, show the familiar splitting of the discrete states (also indicated by white dots). In figure 4.7d an ARRES measurement on exfoliated graphite is presented. In the energy range where we observe the discrete states in multilayer graphene, graphite has a continuous spectrum of high-transmission states (i.e. low reflectance).
∗In literature on SiC this is often referred to as mono, double and triple layer graphene.
There the layer closest to the substrate is not counted because it has different properties than the other layers because it strongly couples to the substrate. We do count this layer to keep the layer count in this section consistent with the rest of the thesis.
4.4. Unoccupied band structure; ARRES 81
c b
500 nm 4.7 eV
2 4 6 8 10
0 10 20 30
I (a.u.)
2 3 4
e
K
(0,0)
M Γ
f
g
(0,0)
(0,0)
a
E0 (eV)
k||
d
k
E
⊥
2
k
⊥+ k2
2m π 8
h2
=
Figure 4.6: (a)LEEM image (acquired at a landing energy of 4.7 eV) of two (bright), three (darker) and four (darkest) graphene layers grown on SiC. (b) LEEM-reflection spectra, for the regions with two (blue), three (green) and four (red) graphene layers.
The curves have been rotated compared to the spectra in the other figures, and are shifted in intensity for clarity. The data are collected from the positions in (a). (c) Schematic side view of SiC covered with bilayer and trilayer graphene (silicon atoms are shown in yellow, carbon atoms in grey). When graphene is grown on SiC the first layer is an electrically insulating buffer layer that resides between SiC and the second (the first true) graphene layer. (d) Sketch of the experiment: In contrast to conventional normal incidence illumination (left), we introduce an in-plane momentum ~kkof the incident electrons by shifting the position of the electron beam in the diffraction plane (right). The electron energy is related to kkand the out-of-plane momentum
~k⊥via the vacuum dispersion relation. It determines the angle of incidence, which is equal to the angle of reflection, as well as the parabolic electron trajectories. (e) Inspection of the diffraction pattern allows us to quantify kk. Here, the untilted case of kk = 0is shown where the specular spot resides at the Γ-point in the center of the Brillouin zone (red hexagon). The dotted circle indicates where the aperture will be placed to detect only specularly reflected electrons (bright-field LEEM). (f) and (g)LEED in the tilted cases, the (0, 0) spot is shifted towards the M-point and to the K-point, respectively. Scale bars in e–g correspond to 1Å−1.
a 65 650
Intensity (a.u.)
100 160 260 400
0.0 0.3 0.6 0.9 1.2 1.5 1.5
k|| (Å–1) 1.2 0.9 0.6 0.3 0
5 10 15 20 25 30
K M
k|| (Å –1) 0
5 10 15 20 25
30 M Γ K
0.0 0.3 0.6 0.9 1.2 1.5 1.5 1.2 0.9 0.6 0.3
b 65 650
Intensity (a.u.)
100 160 260 400
k|| (Å–1) 0
5 10 15 20 25 30
K
M Γ
0.0 0.3 0.6 0.9 1.2 1.5 1.5 1.2 0.9 0.6 0.3
c 65 650
Intensity (a.u.)
100 160 260 400
0 2 2 5
Intensity 6 (a.u.)
100 140 180
k|| (Å–1) 0
5 10 15 20 25 30
E0)Ve(E0)Ve(
E0)Ve(E0)Ve(
K
M Γ
0.0 0.3 0.6 0.9 1.2 1.5 1.5 1.2 0.9 0.6 0.3
d
Γ
Figure 4.7: (a)ARRES measurement on double layer graphene, which is a two- dimensional representation of IV-curves for different in-plane momenta ~kk. The minimum in the LEEM-reflection spectra of a region with two graphene layers (nar- row blue band near the bottom) shifts to higher energies for non-zero kk. (b) Similar behaviour is observed for the minima (two blue bands) in a region with three graphene layers and in (c) with four layers. For all figures, kk is varied from M to Γ to K.
The data at the Γ-point are the LEEM-reflection spectra in figure 4.6b. (d) ARRES measurement on an exfoliated graphite flake showing very similar global behaviour to (a)–(c), but now the discrete bands in a-c have merged to one continuous band in the out-of-plane direction.
4.4. Unoccupied band structure; ARRES 83
(a)
0 2 2 5
Intensity 6 (a.u.)
100 140 180
k|| (Å–1) 0
5 10 15 20 25 30
E0)Ve(
K
M Γ
0.0 0.3 0.6 0.9 1.2 1.5 1.5 1.2 0.9 0.6 0.3
A
(b)
Figure 4.8: (a)Graphite band structure calculated by Hibino et al. [1]. N.b. the energy is measured with respect to the Fermi level in this figure. This figure is from [1]. (b) Graphite ARRES measurement from figure 4.7 with the band edges from (a) as an overlay (mirrored and stretched to the appropriate energy and momentum scale). The band edges in the Γ-K and Γ-M directions, indicated in red in (a), correspond to borders between high and low reflectivity in the graphite ARRES measurement. The Γ-A direction are also shown in the yellow-bordered inset in (b). This shows that only bands that have a dispersion in the out-of-plane direction correspond to features in the ARRES measurement.
When the electron beam is tilted such that the incident electron beam has an in-plane momentum component, the energy of the high-transmission states continuously shifts (the energies at which the resonances occur are again indicated by white dots). The ARRES measurements show that these high- transmission states with in-plane momentum have a parabolic/cosine-like dispersion in the Γ-K and Γ-M directions. Thus, multilayer graphene has discrete states in the out-of-plane (Γ-A) direction and a continuous spectrum in the in-plane direction. This is a consequence of the quasi-two-dimensional nature of multilayer graphene.
In figure 4.8a we show the band structure of graphite, calculated by Hibino et al. [1]. The energy is determined with respect to the Fermi level in this figure.
On the left we see the bands in the Γ-K and Γ-M-directions. If we compare this calculation to the ARRES data for graphite (figure 4.8b), we see that the graphite high-transmission band in ARRES corresponds to the band of which the edges are coloured red in figure 4.8a. Interestingly, the other bands in the energy region are not observed in electron reflection experiments. These
’invisible’ bands all have a flat dispersion in the out-of-plane Γ-A direction.
This can clearly be seen from the yellow-bordered inset in figure 4.8b. The fact that these bands are flat indicates that the states in those bands are localized in the out-of-plane-direction [11]. Incident electrons will therefore not couple into these states [12]. The red band has a strong dispersion in the Γ-A direction.
This is the so-called graphite interlayer band, first described by Posternak [13].
4.5 Conclusions and outlook
In this chapter we have presented LEE reflection spectra of freestanding (multi- layer) graphene. In the double layer graphene spectra oscillations are observed at ∼ 2 eV, ∼ 18 eV and higher energies. In multilayer graphene these split up in multiple resonances around the same energy, only observed around 2 eV.
We show that these are a consequence of unbound electronic interlayer high- transmission states and we have developed a novel technique, called ARRES, to determine the dispersion of these states. Moreover, for the first time, we present LEE transmission spectra. The spectrum of double-layer graphene shows a maximum that coincides with the minimum in the reflection spectrum.
This confirms our earlier expectations due to the toy models in chapter 1 and
4.5. Conclusions and outlook 85
the work of Feenstra et al. We directly probe the high-transmission states. In- terestingly, the reflection and transmission spectra are not exactly each other’s complement and the resonances are qualitatively similar to those found in the toy model that takes loss processes into account. In the next chapter, we show that these loss processes are a consequence of inelastic processes. Inelastic effects play an important role in the transmissivity and reflectivity of LEE.
This effect becomes more pronounced at higher energies.
In the future, it will be worthwhile to improve the eV-TEM design with deflec- tors to control the incident in-plane momentum of the incident electrons. This would allow for ARRES measurements in transmission (i.e. ARTES). Moreover, as we will show in the next chapter, this would allow us to compare the ARRES and ARTES spectra to study the effects on inelastic losses on the unoccupied band structure.
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