# Modeling Field Electron Emission from a Flat Au (100) Surface with Density-Functional Theory

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. DFT Calculation

#### 2.2. DFT Results Post-Processing

#### 2.3. Transmission Coefficient

#### 2.4. Emission Current Densities

#### 2.5. Summary of Numerical Techniques

## 3. Results

#### 3.1. Transmission Coefficient and Pre-Factor Behavior

#### 3.2. Correction Factor to the First-Order Taylor Expansion

#### 3.3. Fowler–Nordheim Plot

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## Appendix B

## Appendix C

**Figure A1.**Decay width as a function of the applied field for four PEs. The dashed red line at 18.92 GV/m represents the reference field ${F}_{\mathrm{R}}$ for the SN PE.

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**Figure 2.**One-dimensional KS PE with a suboptimal k-point grid and a smaller planewave energy cutoff for the purpose of XC selection, using the PBE and PW91 XC functionals. The PE profiles are plotted along the normal direction through the surface, from the center of the slab to the center of the vacuum separating neighboring slabs. Periodic oscillations below $z\approx $30 Bohr radii (the atomic unit of length, $1\mathrm{a}.\mathrm{u}.=1\mathrm{Bohr}\mathrm{radius}\approx 52.92\mathrm{pm}$) correspond to each layer of atoms inside the slab. Note that the PBE profile can be optimized by increasing the size of k-point grid and planewave energy cutoff and applying a Gaussian filter.

**Figure 3.**Longitudinal profiles of the induced electron density ${\mathsf{\rho}}_{induced}$ at fields of 100 MV/m, 300 MV/m, 700 MV/m, and 1 GV/m. The center of the slab is at 0 Bohr radii, and the solid green dot represents the position of the outermost atom, which is fixed for all the fields presented. The dashed orange lines show the centroid locations of the induced electron density, which are 30.40 Bohr radii (F = 100 MV/m), 30.49 Bohr radii (F = 300 MV/m), 30.34 Bohr radii (F = 700 MV/m), and 30.40 Bohr radii (F = 1 GV/m).

**Figure 4.**The one-dimensional KS, IC, ET, and SN PE profiles at zero field from the center of the slab to the center of the vacuum.

**Figure 5.**Ratio of TM transmission coefficient to that of DOP853 for all PE profiles under consideration.

**Figure 7.**Transmission coefficient obtained from the TM method. The dashed red line at 18.92 GV/m represents the reference field ${F}_{\mathrm{R}}$ for the SN PE.

**Figure 8.**Effective pre-factor in the Landau and Lifschitz formula. The dashed gray line at 3 GV/m marks the region of numerical instability to its left. The dashed red line at 18.92 GV/m represents the reference field ${F}_{\mathrm{R}}$ for the SN PE.

**Figure 9.**Correction factor arising from the first-order Taylor expansion. The dashed red line at 18.92 GV/m represents the reference field ${F}_{\mathrm{R}}$ for the SN PE.

**Figure 10.**Fowler–Nordheim plot of four PEs. $F$ denotes the field and $J$ the emission current density. The horizontal axis is the inverse of the field. The vertical axis is on a logarithmic scale. The dashed red line at ${\left(18.92\mathrm{GV}/\mathrm{m}\right)}^{-1}$ represents the inverse of the reference field $1/{F}_{\mathrm{R}}$ for the SN PE.

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**MDPI and ACS Style**

Li, Y.; Mann, J.; Rosenzweig, J.
Modeling Field Electron Emission from a Flat Au (100) Surface with Density-Functional Theory. *Instruments* **2023**, *7*, 47.
https://doi.org/10.3390/instruments7040047

**AMA Style**

Li Y, Mann J, Rosenzweig J.
Modeling Field Electron Emission from a Flat Au (100) Surface with Density-Functional Theory. *Instruments*. 2023; 7(4):47.
https://doi.org/10.3390/instruments7040047

**Chicago/Turabian Style**

Li, Yiming, Joshua Mann, and James Rosenzweig.
2023. "Modeling Field Electron Emission from a Flat Au (100) Surface with Density-Functional Theory" *Instruments* 7, no. 4: 47.
https://doi.org/10.3390/instruments7040047