Quantitative secondary electron imaging for work function extraction at atomic level and layer identification of graphene

Two-dimensional (2D) materials usually have a layer-dependent work function, which require fast and accurate detection for the evaluation of their device performance. A detection technique with high throughput and high spatial resolution has not yet been explored. Using a scanning electron microscope, we have developed and implemented a quantitative analytical technique which allows effective extraction of the work function of graphene. This technique uses the secondary electron contrast and has nanometre-resolved layer information. The measurement of few-layer graphene flakes shows the variation of work function between graphene layers with a precision of less than 10 meV. It is expected that this technique will prove extremely useful for researchers in a broad range of fields due to its revolutionary throughput and accuracy.


Optical and secondary electron (SE) visibility of graphene on metal
substrates Figure S1.1 shows optical and SE images of graphene on two different metal substrates of Cu and Ni. The pristine graphene flakes were exfoliated onto a Si substrate with 285 nm thickness of SiO 2 (SiO 2 /Si), which exhibited a visible optical contrast that can be observed under an optical microscope (Figures S1.1a and S1.1d). After being transferred onto metal substrates, their optical contrast turned to be much lower than those on SiO 2 /Si substrate. For few-layer graphene flakes on both the Cu ( Figure S1.1b) and Ni ( Figure S1.1e) substrates, they are almost invisible under an optical microscope.
On the contrary, the SE images taken by a 5 keV electron beam still show a visible contrast that the few-layer graphene flakes can be clearly observed (Figures S1.1d and S1.1f). The measured SE contrast for monolayer graphene on these two substrates were (7 ± 1) % for Cu and (9 ± 2) %, which are quite close to the value of (10±2) % for monolayer graphene on Au shown in Figure 1a.  Figure S2.1a shows a schematic of light reflection from a graphene sheet on a single layer structured substrate (e.g. metals). The reflected light from graphene surface has two contributions: one is the direct reflection of incident light on air/graphene interface; the other is that the incident light first refracts into graphene layer, then reflect at the graphene/substrate interface and finally refracts back into air. Assuming the incident light is perpendicular to the graphene plane, the total reflected light intensity that is described by Fresnel's raw can be written is:

Calculation of graphene optical contrast on a supported substrate
are the relative indices of refraction. n 0 , n 1 and n 2 are the reflective index of air, graphene and substrate respectively. Φ 1 = 2 1 1 / is the phase shift due to changes in the optical path within graphene layer. For the more complex graphene/SiO 2 /Si trilayer structures, there will be an additional light reflection at the SiO 2 /Si interface.
Therefore the total reflected light from graphene surface is given by: , n 2 and n 3 is the reflective index of SiO 2 and Si layer respectively. Φ 2 = 2 2 2 / is the phase shift due to changes in the optical path within SiO 2 layer, d 2 is the thickness of SiO 2 layer. The optical contrast C is defined as the relative intensity of reflected light in the presence ( 1 ≠ 1) and absence ( 1 = 0 = 1) of graphene, The monolayer graphene has a layer thickness of d 1 = 0.335 nm. It has been reported that the few-layer graphene has the same reflective index as the bulk graphite 1

Calculation of graphene SE contrast on a supported substrate
In this section, we will present a physical model to calculate the SE contrast of graphene on a supported substrate, and discuss its dependence with beam energy and layer thickness. The irradiation of primary electrons (PEs) and generation of SEs are shown in Figure S3.1. A few-layer graphene flake is placed on the surface of a bulk substrate and irradiated by a high energy PE beam. The PEs can penetrate through the thin graphene layers and goes deeply into the substrate before they lose energy. SEs and back-scattered electrons (BSEs) will first be produced within the graphene layer during the PE penetration process, and escape from the graphene surface. The substrate can also generate SEs and BSEs. These electrons will diffuse into the thin graphene layer and escape from the graphene surface. The attenuation of these electrons might also excite SEs within the thin graphene layer by losing energies. The escaped SEs and BSEs might hit the chamber and generate a background SE signal. The total signal that is collected by a SE detector will contain the contribution from all these generated electrons.
In our experiments, we mainly focus on the SE emissions from thin graphene flakes.
The measured flakes usually have the layer number N of ≤ 10 . We can make reasonable assumptions to ignore the signals from some parts of generated electrons in the total collected signal. Firstly, we ignore the energy loss of the PE beam when penetrating through the graphene layer. The Bethe theory estimate that the stopping power of a 5 keV electron in graphite is approximately 7 eV/nm 2 . Therefore the energy loss through a few-layer graphene ( ≤ 10) is less than 1 %. The energy and intensity of the beam that enters the substrate can be regarded as the same as the primary incident beam. Secondly, we ignore the contribution of all BSEs. BSEs generated in graphene Page 8 of 30 can be ignored due to its ultralow yield. Our simulation (using Casino v 2.48) shows that even for the 10 layer graphene, the BSE yield is below 0.001 at about 5 keV.
Although the BSE yield of substrate cannot be ignored (e.g. about 0.1 for the SiO 2 /Si substrate at a 5 keV beam energy), however, the through-the-lens (in-lens) SE detector only collects electrons with small emission angles. The maximum collection angle in our experiment is approximate 26 (working distance 4 mm, lens hole diameter 4mm),therefore the collected BSEs is < 0.02 % and can be ignored. Furthermore, we can ignore the SEs excited by substrate generated SEs. This is because the minimal ionization energy of graphite is 11.26 eV, while from the SE spectrum over 80 % of the attenuated SEs has a low energy below 10 eV. Even a SE is excited, the substrate SEs will lose energy and might not escape from the graphene surface. We estimated the contribution of background signal to be less than 10 % and it can also be ignored.
Therefore, the collected SE current intensity from a graphene surface( ) contains signals that mainly come from: (a) SEs generated by the PE beam (intensity of: where is the current of is the current of the PE beam, ( ) is the SE yield of the thin graphene with layer number of N). (b) SEs generated by substrate BSEs (intensity of: We assume all the escaped SEs can be collected efficiently by the SE detector. The total intensities from graphene ( ) and substrate ( ) surface are given by: The graphene SE contrast is defined as: We first discuss the relationship of monolayer SE contrast with the beam energy, as we presented in Figure 1c. For monolayer graphene we have N=1. We ignore the SE attenuation in the monolayer graphene (i.e. − ≅ 1) The SE contrast becomes: We can obtain the ratio of � � � � from experiment measurements (see next section for the discussion of SE yield). η s can be computed from the simulation using Casino software (version 2.48). The energy distribution of SEs can also be obtained, then β can be calculated to have a value ~ 1.8 for high energy electrons (> 2 keV). We calculated α to be ~ 0.3 by assuming the SEs overcome the surface affinity to escape from the SiO 2 surface and into the graphene/SiO 2 gap first, then penetrate into graphene. It is an underestimated value because at such a small gap the SEs might directly diffuse from the substrate into graphene without energy loss. We found that larger value of α=0.45 to fits the results better in Figure 3c. For graphene on a metal substrate (e.g. Au), will be a constant because the obey the same SE energy distribution. Our simulation also shows that backscattering coefficient of Au doesn't greatly change with beam Page 10 of 30 energy (always around 44% at energy > 1 keV). This will result in a constant contrast at different beam energies.
We then discuss the layer dependence of graphene SE contrast. From the measured SE spectra in Figure 4 we can estimate the most probable energy of escaped SEs is ~ 1.5 eV. SEs in graphene have to overcome a surface potential barrier (i.e. work function of ~4.3 eV). Therefore the mean energy of escaped SEs in graphene can be estimated to be ~6 eV. The IMFP value for a 6 eV electron can be roughly estimated by the Seah and Dench model 3 Where is the thickness of graphene monolayer. The model gives a value of ~6 nm.
This value is much larger than the graphene layer thickness. Therefore we use a linear approximation of: 3) can be written as: Here ( ) can be computed by the integration of the graphene SE energy distribution: Where Φ( ) is the work function for graphene with thickness of N layers. We Where and are the kinetic energies of SEs in the specimen and in vacuum, and are the corresponding incident and refracted angles. We ignore the energy loss for a SE to diffuse from the SiO 2 substrate into graphene, therefore the energy that a SE need to overcome and escape from graphene surface is Φ − χ, where χ is the affinity of SiO 2 layer (χ ≅ 1eV). The maximum escape angle for a SE with energy is: , can be computed as: The difference of values between Eq.(3.6) and Eq.(3.10) shows the deviation from linear contrast decrease that we observed.
When → ∞, Eq. (3.6) corresponds to the SE contrast of a bulk graphite. We assume the BSE yield doesn't change, Eq. (3.6) then becomes:

Evaluation of graphene SE yield
We used a simple method to evaluate the SE yield of the free-standing graphene. Figure S4.1a shows a SEM image of a monolayer graphene on a Si substrate with etched holes (diameter ~ 2 μm, depth > 10 μm). Some holes were covered by the freestanding graphene. We treat the uncovered hole as an ideal Faraday cup because its very deep and very few of SEs and BSEs can escape from the hole. Therefore the primary beam current can be obtained when the hole is irradiated by a primary beam. The current measured from a freestanding graphene will become smaller because the secondary electrons and back-scattered particles will be generated during the beamgraphene interaction process (Fig. S4.1b). Therefore we can obtain the total electron emission yield of a freestanding graphene, which contains contributions from both SE emission ( ) and BSE emission ( ), and is given by: The casino simulation presented in Figure S4.1c shows a low BSE yield below 0.5 % for the e-beam energy higher than 1 keV, which can be ignored. The extracted SE yield is shown as black squares in Figure S4.1d, which has a high value of ~120 % at the low beam energy of ~0.2 keV, and then decreases as beam energy increases. It becomes <10 % when the beam energy is higher than 5 keV. At the same beam energy, the SE yield of SiO 2 surface is around 1 6 . Therefore the SEs generated in the substrate and diffuse into graphene dominate the SE emission. For the HIM irradiation process, the back-scattered ion coefficient can be calculated by the SRIM simulation, which shows a much smaller value (10 -5 at the 30 keV He + beam energy) and can be ignored. So the total yield we measured could be regarded as the SE yield. The result is shown as red Page 14 of 30 circles in Figure S4.1d. The SE yield is much higher than that in SEM, and slightly increases with the increase of He + beam energy.
For the SE yields of the freestanding graphene and substrate that are used in Equation (1)

Raman spectrum
The optical image of imaged few-layer graphene flakes is shown in Figures S5.1a.
The corresponding optical contrast of the red (R), green (G) and blue (B) channels are defined by Eq. (2.3) calculated and presented in Figure S5.1b. . The optical contrast for a N-layer graphene could be fitted by a parabolic relationship 7 . For the green channel with best contrast (~ 5% for monolayer graphene), the relationship is: Therefore the optical contrast could determine the thickness information of few layer graphene. We also used Raman spectrum to confirm the thickness information of few-layer graphene, as shown in Figure S5c. The Raman spectrum of a monolayer graphene has a different intensity ratio of G peak (~1580 cm -1 ) to 2D peak (~2680 cm -1 ) comparing with that of multilayer graphene 8 . However, it is difficult to distinguish the different thicker graphene layers.

Optimisation of the SE imaging conditions in SEM
The influence of primary beam energy and substrate to the SE contrast have been discussed in the manuscript. Here we investigate the other parameters that might have potential influence. All the experiments were done in the Zeiss Supra SEM.
We first varied the working distance at a constant beam energy of 5 keV in the Zeiss Supra SEM. The SE intensity variation of both substrate and different graphene layers is shown in Figure S6.1a. As the working distance increases from 3 mm to 7 mm, the SE intensities of substrate and graphene layers increase first then decrease. The maximum intensity is determined as a working distance between 4 to 5 mm. The  almost as a constant. Therefore the graphene work function extraction based on our contrast model is not sensitive to the working distance, i.e. the SE collection efficiency.
However, we still imaged all graphene flakes at the optimized condition, so that SEs could be collected quite efficiently.
We also investigated the influence of the other imaging parameters, such as scan speed and beam current. Figure S6 Figure S6.3b. The beam current is controlled by the aperture, which varies from 7 μm to 30 μm. The graphene SE contrast also slightly decreases as the beam current increases. Therefore the scan speed and beam current will not greatly influence SE contrast.
As a conclusion, the working distance will strongly affects the SE contrast while the scan speed and beam current only slightly changes to the SE contrast. In our experiment, we select the working distance to be 4 mm to obtain a best contrast.

Sample Layer Contrast
Zeiss Supra FEI Strata       Figure S10.1a shows the SEM image of a freestanding monolayer graphene (marked by the red arrow) that is used for SE spectrum measurement. The data we directly obtained from SE energy filtering measurement is a grid voltage bias versus SE intensity curve. Figure S10.1b shows such a curve obtained from a freestanding graphene. The intensity starts to decrease rapidly at an onset point of

Measurements of graphene SE spectra
indicating SEs starts to be filtered. The relationship between filtered SE energy and grid voltage is = − ( − ) . By converting the x-axis in Figure   S10.1b to SE energy, and differentiating the filtered SE intensity, the SE spectrum describes the proportion of SEs with energy that can escape from the surface, is the BSE coefficient.
We also measured the SE spectra of graphene in HIM. Figure S10.3a shows the SE spectra of a freestanding monolayer graphene in HIM. Figure