# Electronic Processes at the Carbon-Covered (100) Collector Tungsten Surface

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## Abstract

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**E**directed away from the surface. Our aim is to answer the question of an increased penetrability of electrons at the collector side of a nanometric tunnel diode when covered by carbon atoms, a purely quantum mechanical effect related to the value of the workfunction Φ. To obtain Φ at a non-zero electric field we have extrapolated back to the electrical surface the straight line representing the linear increase in the potential energy with distance outside the metal-vacuum interface. We have found that under the presence of

**E**the workfunction Φ = E

_{vac}− E

_{F}of the (100) pure tungsten surface has a minor dependence on

**E**. However, the carbon-covered tungsten (100) surface workfunction Φ(C − W) has a stronger

**E**dependence. Φ(C − W) decreases continuously with the electric field. This decrease is ΔΦ = 0.08 eV when

**E**= 1 V/nm. This ΔΦ is explained by our calculated changes with electric field of the electronic density of both pure and carbon-covered tungsten. The observed phenomena may be relevant to other surfaces of carbon-covered tungsten and may explain the reported collector dependence of current in Scanning Field Emission Microscopy.

## 1. Introduction

**E**-dependent. Explicit numerical calculations by Jensen [11] showed that the field contribution may be several tenths of an eV.

## 2. Method

**E**is pushing the electrons into the surface, were allowed to relax although no changes were recorded beyond the second layer. Note that due to the periodic boundary conditions, a triangular well is formed in front of the charge layer facing the back surface where electrons are pulled out of the metal. The energy cut-off for the expansion of the wavefunction was 317 eV and the condition for convergence of lattice geometry was that the change in the cohesive forces was less than 0.01 eV/A

^{0}. Purely electronic iterations were terminated when the difference in energy was less than 10

^{−4}eV. Convergence of the DFT-VASP calculations are in general difficult as other workers have noted [17] and it is achieved by starting from zero electric field and then increasing it in very small steps of 0.1 V/nm. We have reached a maximum

**E**of 1 V/nm. Unfortunately, we were not able to go to higher fields, but it transpired that this was enough to draw our conclusions. We note that when convergence has been achieved, a minute electron density corresponding to 1/100 of an electron remains in the triangular well in front of the back charge layer and a corresponding charge is missing from the back tungsten surface, see Figure 1 and insets there. This charge density is placed back on the back surface of tungsten and the effects of the transfer are taken into account by simple electrostatics. This refinement does not affect the front surface of the metal under study where the electric field is pushing the electrons into the surface and where our calculations are focused.

## 3. Results-Discussion

_{vac}− E

_{F)}= 3.98 eV. This value is smaller than the experimental value of 4.3–4.4 eV but this is mostly due to the Perdue exchange and correlation (XC) potential used in the present version of VASP. For the variation of the calculated tungsten workfunctions of most tungsten surfaces with respect to the XC functional used see [18]. The corresponding band diagram of the carbon covered tungsten is shown in Figure 2b From that diagram, the result comes out as (E

_{vac}− E

_{F}) = 3.93 eV, i.e., 0.05 eV lower. The previous comments of Figure 2a, referring to the XC functional, also apply here.

**E**= 1 V/nm. Instead the energy at every point outside the W (100) surface increases linearly in accordance with the applied electric field. The answer to this problem can be found in the work of Lang and Kohn [19] and Forbes [20] on the definition of the electrical surface. We remind the reader that this is the surface from which the electric field begins to act and it was shown to be the centroid of the induced surface charge. This is usually taken to be half the interatomic distance away from the surface of the outer nuclei. We therefore extrapolate the straight line representing the increase in potential energy outside the investigated surface back towards this surface and define E

_{vac}as the value of the VASP energy at the electrical surface, see Figure 3a,b.

**E**= 1 V/nm. However, when we examine the corresponding figure of the carbon-covered tungsten surface, Figure 3b, we observe a decreased workfunction Φ(C − W) = 3.9 eV. This gives a change ΔΦ = Φ(W) − Φ(C − W) = 0.08 eV at

**E**= 1 V/nm. We have verified that for values of the electric field up to the value of ε = 1 V/nm this change was proportional to the field. We expect therefore that at the more usual values of

**E**= 4–5 V/nm at which field emission is performed the expected ΔΦ would be around 0.4 eV. This would be a substantial change of the workfunction of the carbon-covered tungsten collector surface by the field and it would affect exponentially any narrow gap diode. Bearing in mind the underestimates that the Perdue potential is producing of the workfunctions, ΔΦ might actually be higher.

**E**= 1 V/nm) − ρ(z,

**E**= 0). The positions of the atoms are also shown in this figure. The red thin lines denote the atoms of pure tungsten. The rightmost dashed thicker black line denotes the carbon atoms of carbon-covered tungsten while the thinner black dashed lines denote the positions of the relaxed tungsten atoms of carbon-covered tungsten. In the case of pure tungsten, there is an accumulation of charge density (red line) just outside the plane of tungsten nuclei and also a depletion layer immediately further out from the accumulation one. Obviously the incoming electrons have “squeezed” the host electronic density into the metal, note that this diagram gives the change in electronic density. A similar “squeezing” occurs in the case of the carbon-covered tungsten (black line), only in this case the “squeezing” is more intense and its accumulation layer has moved into the metal, i.e., to the left of the surface nuclei. Therefore, the incoming electrons will find it easier to enter the metal since they will be penetrating a less charged region compared to pure tungsten. Hence, a reduction in the workfunction should be expected as it does. In a more traditional way, one can view this process as a reduction in surface dipole moment.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Laterally averaged electron density of seven layers of pure tungsten under an electric field of 1 V/nm. The electric field is applied so that it pushes in the electrons at the right hand side (RHS) surface. Near z = 0 a minute charge remains (see LHS inset) which does not affect the RHS surface. The RHS inset shows the form of the potential used by VASP. In all subsequent figures, the applied field is in the same direction.

**Figure 2.**(

**a**) Laterally averaged electron potential of pure tungsten with no applied electric field. The workfunction is 3.98 eV. The Fermi level is denoted by a red line as in all subsequent figures. (

**b**) Laterally averaged electron potential of carbon-covered tungsten with no applied electric field. The workfunction is 3.93 eV.

**Figure 3.**(

**a**) Laterally averaged electron potential of pure tungsten with an applied electric field of 1 V/nm. The workfunction is 3.98 eV the same as with no field. (

**b**) Laterally averaged electron potential of carbon-covered tungsten with an applied electric field of 1 V/nm. The workfunction is 0.08 eV lower that at zero field.

**Figure 4.**Difference in electron density between the cases of zero field and an electric field of 1 V/nm. Red line (triangles) is for pure carbon and black line (circles) for carbon-covered tungsten. See text for details.

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**MDPI and ACS Style**

Gotsis, H.J.; Bacalis, N.C.; Xanthakis, J.P.
Electronic Processes at the Carbon-Covered (100) Collector Tungsten Surface. *Micromachines* **2022**, *13*, 888.
https://doi.org/10.3390/mi13060888

**AMA Style**

Gotsis HJ, Bacalis NC, Xanthakis JP.
Electronic Processes at the Carbon-Covered (100) Collector Tungsten Surface. *Micromachines*. 2022; 13(6):888.
https://doi.org/10.3390/mi13060888

**Chicago/Turabian Style**

Gotsis, Harilaos J., Naoum C. Bacalis, and John P. Xanthakis.
2022. "Electronic Processes at the Carbon-Covered (100) Collector Tungsten Surface" *Micromachines* 13, no. 6: 888.
https://doi.org/10.3390/mi13060888