# Low-Energy Electron Inelastic Mean Free Path of Graphene Measured by a Time-of-Flight Spectrometer

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. SLEEM/ToF Device

#### 2.2. Data Processing

_{L}, and provide the conversion table, i.e., theoretical dependence, t(E), of the ToF on energy, E, of the system with these specific settings. These scripts help to find an optimum setup for each measurement.

_{switch}, trigger pulse sent t

_{pulse}and detection event t

_{detect}. These raw data are converted in true hyper-spectral imaging data of ToF “times” (t

_{detect}− t

_{pulse}). Line and frame switches are identified from regular patterns in the pixel switch signals, which allows one to assign a position to each pixel. Thence, we acquire a collection of ToF values corresponding to a given pixel in a given scan of the selected window, i.e., an image frame. In other words, ToF spectra for each set of the three coordinates—scan/frame no. and two pixel coordinates. We visualize the hyper-spectral data in Figure 4a, displaying density of counts in the time-domain histogram for each pixel collected over all frames. Consider a slice corresponding to a given time bin. We see that the pixels in the slice may contain areas of significantly different densities of detections (higher values are lighter in the image), i.e., the transmissivity, as recorded by the MCP.

_{IMFP}value from energy-loss spectra measured at a given landing energy E

_{L},

#### 2.3. DFT Calculations

## 3. Results

^{6}m/s (experimental value is 1.1 × 10

^{6}m/s [61]). DFT-LDA is known to underestimate the experimental value of Fermi velocity, while the many-body GW simulations often provide better agreement with the experiment [60].

^{−1}, are displayed in Figure 10b. Let us note that positions of both dominant loss-peaks from upon the simple interband transitions estimated from the DFT band structure, Figure 9b, correspond well to the more accurate RPA calculations close to the Г-point (Figure 10b, data for the lowest MT). The positions of maxima of the two main loss features are 4.1 eV and 14.3 eV for q = 0.055 Å

^{−1}in the DFT-EELS spectra. The multiple simulation data-sets displayed in Figure 10b clearly show the dispersion relation of the two loss-peaks.

^{2}+ q

_{E}

^{2}) → dq/q” for q ≠ 0 and 0 if q = 0 ≠ q

_{E}, not including the kinematic restrictions on q. The energy-loss T dependence is contained in q

_{E}that is defined as q

_{E}(T, E

_{L}) = T/(ħ v(E

_{L})), with the velocity v determined from the landing energy E

_{L}. Let us compare the MBPT-EELS intensity from the two segments, collected up to the maximum MT using a weighted sum, with the experimental data (both ZLP and baseline subtracted). Figure 10 shows that both positions of the simulated peaks and their intensity are in a good agreement with the processed experimental spectra. The slight discrepancy is due to the approximations used in the data processing and simulations. Moreover, these simulations support the idea that the splitting of the (π + σ)-plasmon is given by the weighted sum of the momentum-resolved (π + σ)-plasmon intensity over different MTs. This agreement gives us more confidence in the processed experimental data, and we regard them to be ready for the following processing to obtain IMFP.

^{−1}Å

^{−1}, γ = 0.042 eV

^{−1}compare well with the results in Ref. [74], β = 0.0098 eV

^{−1}Å

^{−1}, γ = 0.053 eV

^{−1}. The free-electron plasmon energy of carbon is fixed to E

_{P}= 22.3 eV.

## 4. Discussion

_{plasmon}or energy-loss T

_{plasmon}) and width vary with momentum transfer q, due to the dispersion relation T

_{plasmon}(q), and number of layers n. The π-plasmon range is [4,12] eV and the (π + σ)-plasmon range is [13,30] eV, as shown in Table 1. Furthermore, existing experiments reveal that sample tilt affects the dispersion relation (Figure 4a in Ref. [67]).

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**An overview diagram showing the individual steps of the sample analysis, starting with the sample analysis (

**top**), followed by ToF measurement and processing based on theoretical ray-tracing up to the both ZLP and baseline corrected spectra. Ab initio simulations provide an alternative data input to confirm the measured results. Finally, the ToF measurements at different landing energies allow to determine the energy dependence of the IMFP (

**bottom**).

**Figure 3.**Schematic of the UHV SLEEM/ToF system with the ToF spectrometer installed under the sample for TEs detection (

**a**). Section of a 3D model of sample chamber and ToF spectrometer (

**b**).

**Figure 4.**Transmission image of the graphene sample taken by the MCP detector and the measured ToF spectra. Visualization of the 3D hyperspectral data, the pixel sliced time-domain histogram collected over all frames (

**a**). The non-negative matrix factorization (NNMF) of spectra corresponding to another field of view (

**b**).

**Figure 5.**Heat-map of counts of a 500 eV landing energy measurement (sum over all frames). The high intensity regions are interpreted as holes (

**a**) providing a mask (

**b**); pixels in the white regions are excluded from the cumulative data used to provide the energy-domain histogram.

**Figure 6.**Diffraction patterns of a monolayer graphene obtained with a pixelated detector in Helios SEM; E

_{P}= 30 keV, I

_{P}= 13 pA, 256 × 256 px, camera length = 36.7 mm.

**Figure 7.**As-measured ToF spectrum (blue) of 1 ML graphene layer. Converted energy-loss spectrum (red) contains graphene-related plasmons. Inset: SLEEM image of graphene sheets.

**Figure 8.**ToF energy-domain spectra with similar values of momentum transfer for selected landing energies. The data are divided by both area of the ZLP and width of energy bin.

**Figure 9.**(

**a**) The free-standing monolayer graphene structure produced by VESTA [62] and (

**b**) theoretical DFT results for the total density of states (in states/eV atom) and energy-momentum dispersion relations for the electron bands with some of the important bands labeled accordingly. Vertical arrows (red) indicate transitions that correspond to energy losses of the plasmons. Both DOS and bands are referred to the Fermi energy E

_{F}.

**Figure 10.**Scaled EELS spectra: the processed measurement data at landing energy 320 eV (

**a**) and DFT-calculations (

**b**). The DFT data calculated along the two paths ГM (dash-dotted line) and ГK (dashed line), both segments summed up to momentum transfer q = 0.3 Å

^{−1}(solid line).

**Figure 11.**Effective IMFP calculated from the ToF energy-domain spectra in Figure 10 compared to effective IMFP derived according to TPP-2M and Bethe formulae.

**Table 1.**An overview of various plasmon dispersion relation T(q) (eV) data in the literature, experimental (Exp.) and simulated (Sim). Positions of the plasmon features T(q) (eV) and range of their variations with the measured momentum transfer q (Å

^{−1}), 2nd interval. Dash indicates a value impossible to obtain from the energy-loss range provided in the data.

Reference (Exp.) or (Sim.) | π-Plasmon T(q) (eV), q (Å ^{−1}) | (π + σ)-Plasmon T(q) (eV), q (Å ^{−1}) |
---|---|---|

[65] (Exp.) | [4.9, 6.6], [0.05, 0.38] | -,- |

[66] (Exp.) | [4.9, 8.6], [0.0, 0.7] | -,- |

[67] (Exp.) | [4.0, 12.3], [0.0, 1.56] | [13.5, 30], [0.0, 1.21] |

[63] (Exp.) | [4.8, ≈10], [0.0, 1.4] | [15.3, ≈30], [0.0, 1.4] |

[68] (Sim.) | [4.8, 7.3], [0.03, 1.7] | -,- |

[69] (Sim.) | [≈4, ≈12.5], [≈0.0, 1.4] | [≈14.5, -], [≈0.0, 1.4] |

[63] (Sim.) | [≈4, ≈10], [0.0, 1.3] | [≈14, ≈33], [0.0, 1.3] |

[70] (Sim.) | [4.3, ≈13], [≈0.0, 1.7] | [≈13.9, -], [≈0.0, 1.7] |

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Konvalina, I.; Daniel, B.; Zouhar, M.; Paták, A.; Müllerová, I.; Frank, L.; Piňos, J.; Průcha, L.; Radlička, T.; Werner, W.S.M.;
et al. Low-Energy Electron Inelastic Mean Free Path of Graphene Measured by a Time-of-Flight Spectrometer. *Nanomaterials* **2021**, *11*, 2435.
https://doi.org/10.3390/nano11092435

**AMA Style**

Konvalina I, Daniel B, Zouhar M, Paták A, Müllerová I, Frank L, Piňos J, Průcha L, Radlička T, Werner WSM,
et al. Low-Energy Electron Inelastic Mean Free Path of Graphene Measured by a Time-of-Flight Spectrometer. *Nanomaterials*. 2021; 11(9):2435.
https://doi.org/10.3390/nano11092435

**Chicago/Turabian Style**

Konvalina, Ivo, Benjamin Daniel, Martin Zouhar, Aleš Paták, Ilona Müllerová, Luděk Frank, Jakub Piňos, Lukáš Průcha, Tomáš Radlička, Wolfgang S. M. Werner,
and et al. 2021. "Low-Energy Electron Inelastic Mean Free Path of Graphene Measured by a Time-of-Flight Spectrometer" *Nanomaterials* 11, no. 9: 2435.
https://doi.org/10.3390/nano11092435