# 1D Quantum Simulations of Electron Rescattering with Metallic Nanoblades

^{*}

## Abstract

**:**

## 1. Introduction

^{−2}≈ 6.1 GV/m [10], which is a considerable field considering the fragility of nanotips. In this paper, we introduce the various methods involved with the quantum simulation, including our method for measuring the electron spectrum and the electrostatic/dynamic potential models used. We analyze the results of the simulation and investigate the effects on spectra and yields from changes in peak field amplitude, wavelength, field penetration, and ground state energy. Various alterations to the model and approach are suggested as future work.

## 2. Methods

#### 2.1. Electrostatic Potentials

^{−1}. This atomic potential is centered at $x=0$. The atomic potential and the Jellium potential are combined to produce the electrostatic potential used throughout all simulations, with some universal scaling when performing preliminary total yield calculations.

#### 2.2. Simple Light Field Model

#### 2.3. FDTD-Based Light Field Model

#### 2.4. The Virtual Detector Method for Measuring Emitted Electron Spectra

## 3. Results

^{2}. However, as we reach higher electric fields (above 20 GV/m or $5.3\times {10}^{13}$ W/cm

^{2}), we begin to see interesting hump structures, which arise from expected resonances due to channel-closing when we reach high peak intensities, as seen in atomic ATI [19].

^{2}. This region matches best with the second transition from nonlinearity 2–2.5 to about 1 in Figure 10 (with field penetration). The last regime, being space charge, should not be represented here as the electron-electron interactions were not included in this model. Transitions may also occur as fewer-photon absorption occurs and these slightly excited electrons may more easily tunnel out of the field. Once an electron has absorbed one photon, its effective Keldysh parameter is now 1 at $2.6\times {10}^{13}$ W/cm

^{2}, which is closer to the first transition in the same yield plot. While one cannot attribute the shape of these plots to these regimes without a doubt, this serves as a light interpretation.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ATI | Above Threshold Ionization |

DFT | Density Functional Theory |

FDTD | Finite-Difference Time-Domain |

FFT | Fast Fourier Transform |

FWHM | Full-Width at Half-Max |

HHG | High Harmonic Generation |

TD-DFT | Time-Dependent Density Functional Theory |

TDSE | Time-Dependent Schrödinger Equation |

UCLA | University of California at Los Angeles |

VD | Virtual Detector |

WO | Window Operator |

## References

- Tsujino, S.; Beaud, P.; Kirk, E.; Vogel, T.; Sehr, H.; Gobrecht, J.; Wrulich, A. Ultrafast electron emission from metallic nanotip arrays induced by near infrared femtosecond laser pulses. Appl. Phys. Lett.
**2008**, 92, 193501. [Google Scholar] [CrossRef] [Green Version] - Schenk, M.; Krüger, M.; Hommelhoff, P. Strong-field above-threshold photoemission from sharp metal tips. Phys. Rev. Lett.
**2010**, 105, 257601. [Google Scholar] [CrossRef] [PubMed] - Krüger, M.; Schenk, M.; Hommelhoff, P. Attosecond control of electrons emitted from a nanoscale metal tip. Nature
**2011**, 475, 78. [Google Scholar] [CrossRef] [PubMed] - Freeman, R.; Bucksbaum, P.; Milchberg, H.; Darack, S.; Schumacher, D.; Geusic, M. Above-Threshold Ionization with Subpicosecond Laser Pulses. Phys. Rev. Lett.
**1987**, 59, 1092. [Google Scholar] [CrossRef] [PubMed] - Paulus, G.; Nicklich, W.; Xu, H.; Lambropoulos, P.; Walther, H. Plateau in Above Threshold Ionization Spectra. Phys. Rev. Lett.
**1994**, 72, 2851. [Google Scholar] [CrossRef] [PubMed] - Kruger, M.; Lemell, C.; Wachter, G.; Burgdörfer, J.; Hommelhoff, P. Attosecond physics phenomena at nanometric tips. J. Phys. B
**2018**, 51, 172001. [Google Scholar] [CrossRef] - Li, S.; Champenois, E.; Cryan, J.; Marinelli, A. Coherent Space-Charge to X-ray Up-Conversion in Gas. arXiv
**2018**, arXiv:1806.09650v1. Available online: https://arxiv.org/abs/1806.09650 (accessed on 1 November 2019). - Hermann, R.; Gordon, M. Quantitative comparison of plasmon resonances and field enhancements of near-field optical antennae using FDTD simulations. Opt. Express
**2018**, 26, 27668–27682. [Google Scholar] [CrossRef] [PubMed] - Thomas, S.; Wachter, G.; Lemell, C.; Burgdorfer, J.; Hommelhoff, P. Large optical field enhancement for nanotips with large opening angles. New J. Phys.
**2015**, 17, 063010. [Google Scholar] [CrossRef] [Green Version] - Lennart, S.; Paschen, T.; Hommelhoff, P.; Fennel, T. High-order above-threshold photoemission from nanotips controlled with two-colored laser fields. J. Phys. B.
**2018**, 51, 134001. [Google Scholar] - Parker, N. Numerical Studies of Vortices and Dark Solitons in Atomic Bose-Einstein Condensates. Ph.D. Thesis, University of Durham, Durham, UK, 7 October 2004. [Google Scholar]
- Lang, N.; Kohn, W. Theory of metal surfaces: Charge density and surface energy. Phys. Rev. B
**1970**, 1, 4555. [Google Scholar] [CrossRef] - Jennings, P.; Jones, R.; Weinert, M. Surface barrier for electrons in metals. Phys. Rev. B
**1988**, 37, 6113. [Google Scholar] [CrossRef] [PubMed] - Wachter, G. Simulation of Condensed Matter Dynamics in Strong Femtosecond Laser Pulses. Ph.D. Thesis, Vienna University of Technology, Vienna, Austria, October 2014. [Google Scholar]
- Paschen, T.; Roussel, R.; Heide, C.; Seiffert, L.; Mann, J.; Kruse, B.; Dienstbier, P.; Fennel, T.; Rosenzweig, J.; Hommelhoff, P. Observation of rescattering and space-charge trapping in strong-field photoemission from a macroscopically extruded nanoblade. PRX
**2019**. submitted. [Google Scholar] - Wang, X.; Tian, J.; Eberly, J.H. Virtual detector theory for strong-field atomic ionization. J. Phys. B
**2018**, 51, 084002. [Google Scholar] [CrossRef] - Wang, X.; Tian, J.; Eberly, J.H. Extended Virtual Detector Theory for Strong-Field Atomic Ionization. PRL
**2013**, 110, 243001. [Google Scholar] [CrossRef] [PubMed] - Schafer, K.; Kulander, K. Energy analysis of time-dependent wave functions: Application to above-threshold ionization. Phys. Rev. A
**1990**, 42, 5794. [Google Scholar] [CrossRef] [PubMed] - Paulus, G.; Grasbon, F.; Walther, H.; Kopold, R.; Becker, W. Channel-closing-induced resonances in the above-threshold ionization plateau. Phys. Rev. A
**2001**, 64, 021401(R). [Google Scholar] [CrossRef] - Muller, H.; Linden van den Heuvell, H.; Wiel, M. Experiments on “above-threshold ionization” of atomic hydrogen. Phys. Rev. A
**1986**, 34, 236. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Cross section of the blade system. The laser’s wavenumber is in direction $\mathbf{k}$ (into the page) and the simulation grid is shown as the dashed line, in the $\mathbf{x}$-direction. Rescattering electrons follow a trajectory akin to that shown in blue (although with far less lateral motion if emitted from the blade edge).

**Figure 2.**Electrostatic potential, ${V}_{jellium}\left(x\right)+{V}_{s}\left(x\right)$, with and without external fields applied. The ground state electron is largely bounded to the surface atomic potential.

**Figure 3.**FDTD simulation of our Si-Ti-Au nanoblade. The peak field magnitude (blue) is mostly constant within the metal and follows a near $1/r$ profile outside the metal. The phase (red) includes a $\pi $ phase shift into the metal and some interesting behavior following. Our TDSE (time-dependent Schrödinger equation) simulations do not go far enough into the metal to include these effects as of yet. The first atomic layer is placed at $x=0$ and so the surface of the metal is actually at $x={x}_{im}$ in this plot.

**Figure 4.**Window operator method (red, bold) and virtual detector method (black) compared for a gas electron rescattering test simulation with ${\varphi}_{ce}=0$. The WO (window operator) method ideally performs best in the low energies, where electrons have not had time to leave the simulation. However, the noise level (seen on the right as a flattening out) is fairly high and the WO method involves computationally intense post processing. The VD (virtual detector) method may not have measured all of the low energy electrons as they did not make it to the boundary by the end of the simulation. Regardless, the two spectra overlap closely and the VD method also seems to have a lower noise level, all while requiring just a single FFT (fast Fourier transform) in post processing.

**Figure 5.**A focused view of the electron probability density in space over time. The electron begins at its ground state in the metal towards the left. As the external field is applied, the electron may be excited and/or tunnel out of the metal. Once outside the metal, it may propagate and either return to the metal or escape permanently. Various energy bands may be seen just from this image from the different slopes of electron probability density bunches. This simulation was performed using a 20 GV/m peak enhanced field at 800 nm. Note that the snippet shown here is only a piece of the full simulation—neither spatial nor temporal boundaries are included.

**Figure 6.**A comparison of experimental [15] and simulation data. While both scaling by a constant and exponentiating the data, we arrive at plots that largely agree in the plateau regime. Note that the energy axis is only translated, not scaled, to account for the bias voltage in the experiment.

**Figure 7.**Comparison of spectra among all peak fields simulated (

**Left**) and among mid to low fields (

**Right**). These simulations use the standard potential with 800 nm incident laser pulse, no field penetration.

**Figure 8.**Comparison of standard spectrum (Std) against the modified potential (Mod) (scaled to set ground state to 4.7 eV), penetrating field (Std Pen), and the penetrating field with modified potential (Mod Pen).

**Figure 9.**Electron spectra for varying peak field and wavelength. The semi-classical cutoff is shown as a dashed red line. Variable peak field at 800 nm (

**Left**) shows a good trend of following the classical $10{U}_{p}\propto {E}_{max}^{2}$ cutoff for electron rescatter spectra. Spectra at variable wavelength with peak field of 20 GV/m (

**Right**) also indicates a good general trend to the classical cutoff of $10{U}_{p}\propto {\lambda}^{2}$. This spectral map begins to deviate from the prediction beyond 1600 nm.

**Figure 10.**Simulated electron yield using a modified potential with incident laser wavelength of 800 nm. The electrostatic potential (excluding the external field) was scaled such that the ground state was at –4.7 eV, a much more reasonable value for a metal’s work function. The yields with (

**Right**) and without (

**Left**) field penetration are shown. The individual simulated yields are shown as o’s and the integrated yields (averaged over a 2D Gaussian intensity profile) are shown as x’s. Experiments have shown power laws of about 2.5, tapering to 1 [15]. While we do not see those exact values here, we do see regime transitions. Note that the reported laser intensity is the enhanced intensity as opposed to incident.

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**MDPI and ACS Style**

Mann, J.; Lawler, G.; Rosenzweig, J.
1D Quantum Simulations of Electron Rescattering with Metallic Nanoblades. *Instruments* **2019**, *3*, 59.
https://doi.org/10.3390/instruments3040059

**AMA Style**

Mann J, Lawler G, Rosenzweig J.
1D Quantum Simulations of Electron Rescattering with Metallic Nanoblades. *Instruments*. 2019; 3(4):59.
https://doi.org/10.3390/instruments3040059

**Chicago/Turabian Style**

Mann, Joshua, Gerard Lawler, and James Rosenzweig.
2019. "1D Quantum Simulations of Electron Rescattering with Metallic Nanoblades" *Instruments* 3, no. 4: 59.
https://doi.org/10.3390/instruments3040059