# Negative Effective Mass in Plasmonic Systems

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## Abstract

**:**

## 1. Introduction

## 2. Results and Discussion

#### 2.1. Negative Effective Mass and Plasma Oscillations in Metals

#### 2.2. Negative Mass and Low Frequency Plasmons in 1D Metallic Meso-Structures

_{2}(Li) = 2.4 × 10

^{−9}N/m, k

_{2}(Au) = 1.9 × 10

^{−9}N/m. The optical and acoustical branches of the longitudinal modes propagation in the 1D lattice, depicted in Figure 5, should be elucidated. It should be emphasized that the ensembles of metallic wires, shown schematically in Figure 5, will not demonstrate simultaneously the negative mass (density) and the negative refraction effects [20,21]. This is due to the fact that the negative refraction becomes possible below the plasma frequency ${\omega}_{p}$ [20,21]; contrastingly, the effect of the negative mass in our model emerges when the frequency $\omega $ approaches ${\omega}_{p}$ from above; thus, the creation of material demonstrating the negative density and dielectric constant simultaneously remains challenging. A more comprehensive approach should consider inevitable losses resulting in the decay of plasmons [22], consequently influencing the effect of the negative mass considerably, as discussed in [23].

## 3. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Felbacq, D.; Bouchitté, G. Metamaterials Modelling and Design, Taylor & Francis; Pan Stanford Publishing: Singapore, 2017. [Google Scholar]
- Engheta, N.; Ziolkowski, R.W. Electromagnetic Metamaterials: Physics and Engineering Explorations; IEEE Press: Hoes Lane, NJ, USA, 2006. [Google Scholar]
- Kshetrimayum, R.S. A brief intro to metamaterials. IEEE Potentials
**2004**, 23, 44–46. [Google Scholar] [CrossRef] - Boardman, A.D.; Grimalsky, V.; Kivshar, Y.; Koshevaya, S.; Lapine, M.; Litchinitser, N.; Malnev, V.; Noginov, M.; Rapoport, Y.; Shalaev, V. Active and tunable metamaterials. Laser Photon. Rev.
**2010**, 5, 287–307. [Google Scholar] [CrossRef] - Li, J.; Chan, C.T. Double-negative acoustic metamaterial. Phys. Rev. E
**2004**, 70, 055602. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Grima-Cornish, J.N.; Grima, J.; Attard, D. A Novel Mechanical Metamaterial Exhibiting Auxetic Behavior and Negative Compressibility. Materials
**2019**, 13, 79. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zhang, S.; Yin, L.; Fang, N.X. Focusing Ultrasound with an Acoustic Metamaterial Network. Phys. Rev. Lett.
**2009**, 102, 194301. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chen, H.; Ding, C. Simulated and Experimental Research of Multi-Band Acoustic Metamaterial with a Single Resonant Structure. Materials
**2019**, 12, 3469. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Yang, Z.; Mei, J.; Yang, M.; Chan, N.H.; Sheng, P. Membrane-type acoustic metamaterial with negative dynamic mass. Phys. Rev. Lett.
**2008**, 101, 204301. [Google Scholar] [CrossRef] [PubMed] - Al Sabouni-Zawadzka, A.; Gilewski, W. Smart Metamaterial Based on the Simplex Tensegrity Pattern. Materials
**2018**, 11, 673. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Mei, J.; Ma, G.; Yang, M.; Yang, Z.; Wen, W.; Sheng, P. Dark acoustic metamaterials as super absorbers for low-frequency sound. Nat. Commun.
**2012**, 3, 756. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chan, C.T.; Li, J.; Fung, K.H. On extending the concept of double negativity to acoustic waves. J. Zhejiang Univ. A
**2006**, 7, 24–28. [Google Scholar] [CrossRef] - Huang, H.; Sun, C.; Huang, G. On the negative effective mass density in acoustic metamaterials. Int. J. Eng. Sci.
**2009**, 47, 610–617. [Google Scholar] [CrossRef] - Tonks, L.; Langmuir, I. Oscillations in Ionized Gases. Phys. Rev.
**1929**, 33, 195–210. [Google Scholar] [CrossRef] - Huang, H.H.; Sun, C.T. Wave attenuation mechanism in an acoustic metamaterial with negative effective mass density. New J. Phys.
**2009**, 11, 13003. [Google Scholar] [CrossRef] - Gracià-Salgado, R.; García-Chocano, V.M.; Torrent, D.; Sanchez-Dehesa, J. Negative mass density and ρ-near-zero quasi-two-dimensional metamaterials: Design and applications. Phys. Rev. B
**2013**, 88, 224305. [Google Scholar] [CrossRef] - Yao, S.; Zhou, X.; Hu, G. Investigation of the negative-mass behaviors occurring below a cut-off frequency. New J. Phys.
**2010**, 12, 103025. [Google Scholar] [CrossRef] - Mizutani, U. Introduction to the Electron Theory of Metals; Cambridge University Press (CUP): Cambridge, UK, 2001. [Google Scholar]
- Ashcroft, N.; Mermin, N.D. Solid State Physics; Rinehart & Winston: Holt, NY, USA, 1976. [Google Scholar]
- Pendry, J.; Holden, A.J.; Youngs, I.; Stewart, W.J. Extremely Low Frequency Plasmons in Metallic Mesostructures. Phys. Rev. Lett.
**1996**, 76, 4773–4776. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Veselago, V.; Braginsky, L.; Shklover, V.; Hafner, C. Negative Refractive Index Materials. J. Comput. Theor. Nanosci.
**2006**, 3, 189–218. [Google Scholar] [CrossRef] - Teperik, T.V.; Popov, V.; De Abajo, F.J.G. Radiative decay of plasmons in a metallic nanoshell. Phys. Rev. B
**2004**, 69, 155402. [Google Scholar] [CrossRef] - Henriquez, V.C.; García-Chocano, V.M.; Sanchez-Dehesa, J. Viscothermal Losses in Double-Negative Acoustic Metamaterials. Phys. Rev. Appl.
**2017**, 8, 014029. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**(

**A**) Core with mass ${m}_{2}$ is connected internally through the spring with ${k}_{2}$ to a shell with mass ${m}_{1}$. The system is subjected to the sinusoidal force $F\left(t\right)=\widehat{F}sin\omega t$. (

**B**) Free electrons gas ${m}_{2}$ is embedded into the ionic lattice ${m}_{1}$; ${\omega}_{p}$ is the plasma frequency (the left sketch). The equivalent mechanical scheme of the system (right sketch).

**Figure 2.**The dependence of the dimensionless mass ${m}_{eff}/\left({m}_{1}+{m}_{2}\right)$ on the ratio $\omega /{\omega}_{p}$ is plotted; the red line corresponds to Au; the blue line corresponds to Li.

**Figure 3.**The dependence of the dimensionless effective mass calculated for Li on the $\frac{\omega -{\omega}_{p}}{{\omega}_{p}}=\frac{\mathsf{\Delta}\omega}{{\omega}_{p}}$.

**Figure 4.**The dependence of the dimensionless effective mass calculated for Au on the $\frac{\omega -{\omega}_{p}}{{\omega}_{p}}=\frac{\mathsf{\Delta}\omega}{{\omega}_{p}}$.

**Figure 5.**One-dimensional lattice built of metallic wires 2r connected with springs ${k}_{1}$. The separation between wires is a.

Metal | m_{1} (kg) | m_{2} (kg) | n (m^{−3}) | ω_{p} (Hz) | ${\mathit{k}}_{2}={\mathit{\omega}}_{\mathit{p}}^{2}{\mathit{m}}_{2}\text{}(\mathbf{N}/\mathbf{m})$ |
---|---|---|---|---|---|

Li | 1.17 × 10^{−26} | 9.1 × 10^{−31} | 4.7 × 10^{28} | 1.0 × 10^{16} | 90.0 |

Au | 3.27 × 10^{−25} | 9.1 × 10^{−31} | 5.9 × 10^{28} | 1.3 × 10^{16} | 152.1 |

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**MDPI and ACS Style**

Bormashenko, E.; Legchenkova, I.
Negative Effective Mass in Plasmonic Systems. *Materials* **2020**, *13*, 1890.
https://doi.org/10.3390/ma13081890

**AMA Style**

Bormashenko E, Legchenkova I.
Negative Effective Mass in Plasmonic Systems. *Materials*. 2020; 13(8):1890.
https://doi.org/10.3390/ma13081890

**Chicago/Turabian Style**

Bormashenko, Edward, and Irina Legchenkova.
2020. "Negative Effective Mass in Plasmonic Systems" *Materials* 13, no. 8: 1890.
https://doi.org/10.3390/ma13081890