# Are Current Discontinuities in Molecular Devices Experimentally Observable?

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## Abstract

**:**

## 1. Introduction

## 2. General Equations for $\mathbf{E}$ and $\mathbf{B}$ and Their Radiative Solution

#### 2.1. Radiative Solution in the Dipole Approximation

## 3. Possible Experimental Signatures of a Missing $\mathbf{B}$ and a Radiative ${\mathbf{E}}_{L}$

## 4. Numerical Solutions of the Extended Equations with the Potentials

`NIntegrate`of

`Mathematica`and checking the results by comparison with a Monte Carlo integration in 3D.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Appendix A. Equations in SI Units

## Appendix B. Alternative Proof of the Independence of the EM Fields on the Scalar Source

## References

- Li, C.; Wan, L.; Wei, Y.; Wang, J. Definition of current density in the presence of a non-local potential. Nanotechnology
**2008**, 19, 155401. [Google Scholar] - Zhang, L.; Wang, B.; Wang, J. First-principles calculation of current density in molecular devices. Phys. Rev. B
**2011**, 84, 115412. [Google Scholar] - Lai, L.; Chen, J.; Liu, Q.; Yu, Y. Charge nonconservation of molecular devices in the presence of a nonlocal potential. Phys. Rev. B
**2019**, 100, 125437. [Google Scholar] - Cheng, T.P.; Li, L.F. Gauge Theory of Elementary Particle Physics; Clarendon Press Oxford: Oxford, UK, 1984. [Google Scholar]
- Parameswaran, S.; Grover, T.; Abanin, D.; Pesin, D.; Vishwanath, A. Probing the chiral anomaly with nonlocal transport in three-dimensional topological semimetals. Phys. Rev. X
**2014**, 4, 031035. [Google Scholar] - Ohmura, T. A new formulation on the electromagnetic field. Prog. Theor. Phys.
**1956**, 16, 684–685. [Google Scholar] - Aharonov, Y.; Bohm, D. Further discussion of the role of electromagnetic potentials in the quantum theory. Phys. Rev.
**1963**, 130, 1625. [Google Scholar] - Alicki, R. Generalised electrodynamics. J. Phys. A Math. Gen.
**1978**, 11, 1807. [Google Scholar] - Cornille, P. On the propagation of inhomogeneous waves. J. Phys. D Appl. Phys.
**1990**, 23, 129. [Google Scholar] - Van Vlaenderen, K.; Waser, A. Generalisation of classical electrodynamics to admit a scalar field and longitudinal waves. Hadron. J.
**2001**, 24, 609–628. [Google Scholar] - Jiménez, J.; Maroto, A. Cosmological magnetic fields from inflation in extended electromagnetism. Phys. Rev. D
**2011**, 83, 023514. [Google Scholar] - Hively, L.; Giakos, G. Toward a more complete electrodynamic theory. Int. J. Signal Imaging Syst. Eng.
**2012**, 5, 3–10. [Google Scholar] - Modanese, G. Generalized Maxwell equations and charge conservation censorship. Mod. Phys. Lett. B
**2017**, 31, 1750052. [Google Scholar] - Modanese, G. Electromagnetic coupling of strongly non-local quantum mechanics. Phys. B Condens. Matter
**2017**, 524, 81–84. [Google Scholar] - Arbab, A. Extended electrodynamics and its consequences. Mod. Phys. Lett. B
**2017**, 31, 1750099. [Google Scholar] - Hively, L.; Loebl, A. Classical and extended electrodynamics. Phys. Essays
**2019**, 32, 112–126. [Google Scholar] - Reed, D.; Hively, L. Implications of Gauge-Free Extended Electrodynamics. Symmetry
**2020**, 12, 2110. [Google Scholar] - Cabra, G.; Jensen, A.; Galperin, M. On simulation of local fluxes in molecular junctions. J. Chem. Phys.
**2018**, 148, 204103. [Google Scholar] - Jensen, A.; Garner, M.; Solomon, G. When current does not follow bonds: Current density in saturated molecules. J. Phys. Chem. C
**2019**, 123, 12042–12051. [Google Scholar] - Garner, M.; Jensen, A.; Hyllested, L.; Solomon, G. Helical orbitals and circular currents in linear carbon wires. Chem. Sci.
**2019**, 10, 4598–4608. [Google Scholar] - Garner, M.; Bro-Jørgensen, W.; Solomon, G. Three distinct torsion profiles of electronic transmission through linear carbon wires. J. Phys. Chem. C
**2020**, 124, 18968–18982. [Google Scholar] - Walz, M.; Bagrets, A.; Evers, F. Local current density calculations for molecular films from ab initio. J. Chem. Theory Comput.
**2015**, 11, 5161–5176. [Google Scholar] - Joachim, C.; Gimzewski, J.K.; Aviram, A. Electronics using hybrid-molecular and mono-molecular devices. Nature
**2000**, 408, 541–548. [Google Scholar] - Bachtold, A.; Hadley, P.; Nakanishi, T.; Dekker, C. Logic circuits with carbon nanotube transistors. Science
**2001**, 294, 1317–1320. [Google Scholar] - Nozaki, D.; Schmidt, W. Current density analysis of electron transport through molecular wires in open quantum systems. J. Comput. Chem.
**2017**, 38, 1685–1692. [Google Scholar] - Lenzi, E.; de Oliveira, B.; da Silva, L.; Evangelista, L. Solutions for a Schrödinger equation with a nonlocal term. J. Math. Phys.
**2008**, 49, 032108. [Google Scholar] - Lenzi, E.; De Oliveira, B.; Astrath, N.; Malacarne, L.; Mendes, R.; Baesso, M.; Evangelista, L. Fractional approach, quantum statistics, and non-crystalline solids at very low temperatures. Eur. Phys. J. Condens. Matter Complex Syst.
**2008**, 62, 155–158. [Google Scholar] - Latora, V.; Rapisarda, A.; Ruffo, S. Superdiffusion and out-of-equilibrium chaotic dynamics with many degrees of freedoms. Phys. Rev. Lett.
**1999**, 83, 2104. [Google Scholar] - Caspi, A.; Granek, R.; Elbaum, M. Enhanced diffusion in active intracellular transport. Phys. Rev. Lett.
**2000**, 85, 5655. [Google Scholar] - Chamon, L.; Pereira, D.; Hussein, M.; Ribeiro, M.; Galetti, D. Nonlocal description of the nucleus-nucleus interaction. Phys. Rev. Lett.
**1997**, 79, 5218. [Google Scholar] - Balantekin, A.B.; Beacom, J.F.; Cândido Ribeiro, M.A. Green’s function for nonlocal potentials. J. Phys. G Nucl. Part. Phys.
**1998**, 24, 2087. [Google Scholar] - Laskin, N. Fractional Schrödinger equation. Phys. Rev. E
**2002**, 66, 056108. [Google Scholar] - Wei, Y. Comment on “Fractional quantum mechanics” and “Fractional Schrödinger equation”. Phys. Rev. E
**2016**, 93, 066103. [Google Scholar] - Modanese, G. Time in quantum mechanics and the local non-conservation of the probability current. Mathematics
**2018**, 6, 155. [Google Scholar] - Modanese, G. Design of a test for the electromagnetic coupling of non-local wavefunctions. Results Phys.
**2019**, 12, 1056–1061. [Google Scholar] - Dreyer, C.; Stengel, M.; Vanderbilt, D. Current-density implementation for calculating flexoelectric coefficients. Phys. Rev. B
**2018**, 98, 075153. [Google Scholar] - Modanese, G. High-frequency electromagnetic emission from non-local wavefunctions. Appl. Sci.
**2019**, 9, 1982. [Google Scholar] - Giakos, G.; Ishii, T. Detection of longitudinal electromagnetic fields in air. Microw. Opt. Technol. Lett.
**1993**, 6, 283–287. [Google Scholar] - Monstein, C.; Wesley, J.P. Observation of scalar longitudinal electrodynamic waves. EPL (Europhys. Lett.)
**2002**, 59, 514. [Google Scholar] - Monstein, C.; Wesley, J. Remarks to the Comment by J.R. Bray and M.C. Britton on “Observation of scalar longitudinal electrodynamic waves”. EPL (Europhys. Lett.)
**2004**, 66, 155. [Google Scholar] - Butterworth, E.; Allison, C.; Cavazos, D.; Mullen, F. Longitudinal electromagnetic waves? The Monstein-Wesley experiment reconstructed. J. Sci. Explor
**2013**, 27, 13–23. [Google Scholar] - Umul, Y. Excitation of electromagnetic waves by a discontinuous electric line source. Optik
**2018**, 169, 96–108. [Google Scholar] - Simulik, V.; Gordievich, I.; Zajac, T. Slightly generalized Maxwell system and longitudinal components of solution. J. Phys.
**2019**, 1416, 012033. [Google Scholar] - Wang, J. Time-dependent quantum transport theory from non-equilibrium Green’s function approach. J. Comput. Electron.
**2013**, 12, 343–355. [Google Scholar] - Yu, Y.; Zhan, H.; Wei, Y.; Wang, J. Current-conserving and gauge-invariant quantum ac transport theory in the presence of phonon. Phys. Rev. B
**2014**, 90, 075407. [Google Scholar] - Pohl, V.; Marsoner Steinkasserer, L.; Tremblay, J. Imaging Time-Dependent Electronic Currents through a Graphene-Based Nanojunction. J. Phys. Chem. Lett.
**2019**, 10, 5387–5394. [Google Scholar] - Walz, M.; Wilhelm, J.; Evers, F. Current patterns and orbital magnetism in mesoscopic dc transport. Phys. Rev. Lett.
**2014**, 113, 136602. [Google Scholar]

**Figure 1.**Parameters and location of the Gaussian extra-sources employed in the numerical simulations. (Orange: current sink. Blue: current source).

**Figure 2.**(

**a**) Contributions of the “cloud” of secondary current to the anomalous field B

^{s}for spherical sink and source (see Table 1), in the y range [0–10] 10

^{−7}cm. In the 6D Monte Carlo integration, the 3D region of the y variable has been subdivided into 100

^{3}cells; for each cell, the contribution of sampling points falling into it is displayed. The numerical values on the color scale must be normalized to compute the field, and are therefore not meaningful as absolute value, but their sign and relative values are of interest. In this figure we have y

_{3}= 0, that is, we are observing the cloud on the plane equidistant from source and sink. All contributions are positive, since we are between the sources (compare Figure 3). (

**b**) Here we have y

_{1}= 0, therefore we are observing the cloud in the plane that cuts sink and source. Note that sink and source both give positive contributions on the inner side, and negative contributions on the outer side. From Table 1 we deduce that the total contribution of the region shown in this figure is positive.

**Figure 3.**Same sources and cutting plane as in Figure 2b, but in the y range [10–50$]\xb7{10}^{-7}$ cm. (The central region [0–10] is not sampled because it would give a strong noise, compared to the larger region.) Note that the pattern of Figure 2b is confirmed concerning the inner/outer regions with positive/negative contributions respectively.

**Figure 4.**This figure can be compared with Figure 2a, with the only difference that sink and source are here Gaussian disks/ellypsoids of diameter $D=10\xb7{10}^{-7}$ cm instead of $D=0.5\xb7{10}^{-7}$ cm. (D corresponds to the $\sigma $ of the Gaussian density distribution.) As in Figure 2a, the contributions shown lie on the plane between source and sink, with ${y}_{3}=0$.

**Figure 5.**This figure can be compared with Figure 4, but with source-sink having diameter $D=10\xb7{10}^{-7}$ cm.

**Table 1.**Contributions of the secondary current $\nabla S$, in various regions of the y-space, to the ratio ${B}^{s}/{B}^{0}$. ${B}^{s}$ is the anomalous field generated by the secondary current. ${B}^{0}$ is the Biot-Savart field that the same total current would generate if flowing as a primary local current through the gap of length $2a$. The field is computed at a distance $r={10}^{-4}$ cm from the origin, on the ${x}_{1}$ axis. The shape of the current sink and source is spherical ($\epsilon =D=0.5\xb7{10}^{-7}$ cm). The integration range in ${z}_{1}$ and ${z}_{2}$ is $R=2\xb7{10}^{-7}$ cm, in ${z}_{3}$ is ${R}_{3}=5\xb7{10}^{-7}$ cm. In the y range [800–1100$]\xb7{10}^{-7}$ cm, which contains the point where ${B}^{s}$ is computed, the integration is further split into two parts in order to reduce the noise due to the factor $1/|\mathbf{x}-\mathbf{y}|$. As seen from the last row of the table, the total anomalous field is zero within errors (“missing field” effect).

Range ${\mathit{R}}_{\mathit{y}}$ (Units ${10}^{-7}$) | Contribution to ${\mathit{B}}^{\mathit{s}}/{\mathit{B}}^{0}$ |
---|---|

[0–10] | $0.338$ |

[10–50] | $-0.005$ |

[50–100] | $-0.001$ |

[100–200] | $-0.004$ |

[200–400] | $-0.001$ |

[400–800] | $-0.036\pm 0.003$ |

[800–1100] | $-0.09\pm 0.01$ |

[1100–1600] | $-0.133$ |

[1600–3200] | $-0.046$ |

[3200–6400] | $-0.006$ |

Total | $0.016\pm 0.014$ |

**Table 2.**Same as in Table 1, but for a sink and source with the shape of a disk/ellipsoid ($\epsilon =0.5\xb7{10}^{-7}$ cm, $D=2.5\xb7{10}^{-7}$ cm). The integration range in ${z}_{1}$ and ${z}_{2}$ changes accordingly: here $R=10\xb7{10}^{-7}$ cm. The total anomalous field is again zero within errors; the single contributions differ significantly from the case of spherical sink and source only at small ${R}_{y}$ range. The data for a wider disk ($D=10\xb7{10}^{-7}$ cm, not shown here) reveal a similar behavior: in the first two y ranges we have respectively contributions 0.185 and 0.143, with no substantial differences in the other ranges.

Range ${\mathit{R}}_{\mathit{y}}$ (Units ${10}^{-7}$) | Contribution to ${\mathit{B}}^{\mathit{s}}/{\mathit{B}}^{0}$ |
---|---|

[0–10] | $0.324$ |

[10–50] | $0.009$ |

[50–100] | $-0.000$ |

[100–200] | $-0.003$ |

[200–400] | $-0.014$ |

[400–800] | $-0.042\pm 0.003$ |

[800–1100] | $-0.088\pm 0.005$ |

[1100–1600] | $-0.133$ |

[1600–3200] | $-0.046$ |

[3200–6400] | X |

Total | $0.007+X\pm 0.010$ |

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Minotti, F.; Modanese, G.
Are Current Discontinuities in Molecular Devices Experimentally Observable? *Symmetry* **2021**, *13*, 691.
https://doi.org/10.3390/sym13040691

**AMA Style**

Minotti F, Modanese G.
Are Current Discontinuities in Molecular Devices Experimentally Observable? *Symmetry*. 2021; 13(4):691.
https://doi.org/10.3390/sym13040691

**Chicago/Turabian Style**

Minotti, F., and G. Modanese.
2021. "Are Current Discontinuities in Molecular Devices Experimentally Observable?" *Symmetry* 13, no. 4: 691.
https://doi.org/10.3390/sym13040691