# Impedance Model of Cylindrical Nanowires for Metamaterial Applications

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## Abstract

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## 1. Introduction

- New cylindrical wave impedance approach (impedance defined as $\mathbf{E}/\mathbf{H}$) is used at optical frequency, which uses cylindrical harmonic functions to derive expression for the internal and external impedances of the metallic nanowire.
- Equivalent analytical expression for the scattering (${Q}_{sca}$) and extinction (${Q}_{ext}$) cross-sectional area efficiencies of the nanowire (basic unit cell in metamaterial) is derived in terms of impedances.
- Derived expressions are in terms of length, radius, and dielectric function of the nanowires, thus preserving macroscopic plasmon properties of the metal at optical frequency.

## 2. EM Field Solution to the Scattering, Absorption and Extinction in Circular Cylindrical Nanowire

## 3. Proposed Wave Impedance Model of Metallic Cylindrical Nanowires

#### 3.1. **TM** Mode: Wave Impedance and Cross-Sectional Areas Efficiencies (Scattering, Absorption and Extinction)

**E**and

**H**along $+z$ and $-y$ directions, respectively. This will excite $\mathbf{TM}$ mode in the nanowire. To start with the derivation of impedance, all we need is the relationship of ${E}_{z}^{sca}$ (2) and ${E}_{z}^{abs}$ (3) and we can drive expressions of ${H}_{\varphi}^{sca}$ and ${H}_{\varphi}^{abs}$ using Maxwell’s equation as

#### 3.1.1. Internal Impedance

#### 3.1.2. External Impedance

#### 3.1.3. **TM** Mode: Cross-Sectional Areas Efficiencies Using Impedances

- -
- ${I}_{ext}^{TM}\left({k}_{1}a\right)$: Surface current due to nanowire with property ${k}_{1}$, supporting scattering waves.
- -
- ${I}_{ext}^{TM}\left({k}_{2}a\right)$: Surface current due to nanowire with property ${k}_{2}$, supporting scattering waves.
- -
- ${I}_{int}^{TM}\left({k}_{1}a\right)$: Surface current due to nanowire with property ${k}_{1}$, supporting waves absorption.
- -
- ${I}_{int}^{TM}\left({k}_{2}a\right)$: Surface current due to nanowire with property ${k}_{2}$, supporting waves absorption.
- -
- ${Z}_{\beta}$: The ratio between ${I}_{int}^{TM}\left({k}_{2}a\right)$ and ${I}_{ext}^{TM}\left({k}_{2}a\right)$.

**E**along $+z$ axis. Therefore, from definition of (5) and (6), the scattering cross-sectional area efficiencies using (29) can be written as

#### 3.2. **TE** Mode: Wave Impedance and Cross-Sectional Area Efficiencies (Scattering, Absorption and Extinction)

**E**and

**H**along $+y$ and $+z$ directions, respectively. To start with the derivation of impedance, all we need is the relationship of ${H}_{z}^{sca}$, ${H}_{z}^{abs}$, ${E}_{\varphi}^{sca}$ and ${E}_{\varphi}^{abs}$, which are the $\mathbf{TE}$ components of the scattered and absorbed waves of the electric and magnetic fields respectively. From the preceding sections, the respective $\mathbf{H}$ and $\mathbf{E}$ fields values can be written as follows

#### 3.2.1. Internal and External Admittance

#### 3.2.2. **TE** Mode: Cross-Sectional Areas Efficiencies Using Admittance

**TE**solution, we will follow the analogous approach used in solving

**TM**case. The voltages are defined as follows:

- -
- ${V}_{ext}^{TE}\left({k}_{1}a\right)$: Voltage due to nanowire with property ${k}_{1}$, supporting scattering waves.
- -
- ${V}_{ext}^{TE}\left({k}_{2}a\right)$: Voltage due to nanowire with property ${k}_{2}$, supporting scattering waves.
- -
- ${V}_{int}^{TE}\left({k}_{1}a\right)$: Voltage due to nanowire with property ${k}_{1}$, supporting waves absorption.
- -
- ${V}_{int}^{TE}\left({k}_{2}a\right)$: Voltage due to nanowire with property ${k}_{2}$, supporting waves absorption.
- -
- ${Y}_{\alpha}$: The ratio between ${V}_{int}^{TE}\left({k}_{2}a\right)$ and ${V}_{ext}^{TE}\left({k}_{2}a\right)$.

## 4. Simulation and Results

**TE**and

**TM**modes. In FDTD, the extinction, scattering and absorption cross-section efficiencies are solved using plane wave excitation and defining the separate regions of Total Field (TF) and Scattered Field (SF) for the measurement. The nanowire is placed in TF region, and the two regions are separated by virtual boundary. Absorbing conditions are used and the results of the cross-sectional efficiencies are plotted and compared with proposed impedance method in Figure 7 and Figure 8.

**TE**mode. The red dots show the results of field solver simulation. Please note that the simulation is for a 40 nm radius cylindrical nanowire, with normalized cross-sections values. In

**TE**mode, the resonance peaks appear at 346 nm, 350 nm and 343.5 nm for the extinction, scattering, and absorption cross-sectional areas, respectively. Similarly, for the

**TM**mode the results are plotted in Figure 8. The resonance peaks occur at 462 nm, 439 nm and 480.5 nm for the ${Q}_{ext}$, ${Q}_{sca}$ and ${Q}_{abs}$ respectively. The result shows negligible difference between the proposed impedance model and full-wave field solver simulation.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

E | Electric |

M | Magnetic |

TE | Transverse Electric |

TM | Transverse Magnetic |

FDTD | Finite Difference Time Domain |

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**Figure 1.**A uniform $\mathbf{TM}$/$\mathbf{TE}$ plane wave with $+x$ as direction of propagation in free space is incident on cylindrical nanowire of radius a.

**Figure 2.**Extinction, scattering and absorption cross-sectional efficiencies (dimensionless) plotted using field solution for the excited $\mathbf{TE}$ and $\mathbf{TM}$ modes for a 30 nm (nanometer) radius gold nanowire.

**Figure 3.**Representation of internal impedance of the excited ${\mathbf{TM}}_{n}$ mode in the cylindrical nanowire forming a ladder network.

**Figure 4.**Extinction of a single nanowire. (

**a**) The internal medium property ${k}_{2}=\omega \sqrt{{\u03f5}_{2}{\mu}_{2}}$ is same as external medium property ${k}_{2}$. (

**b**) The material property ${k}_{1}=\omega \sqrt{{\u03f5}_{1}{\mu}_{1}}$ (of cylinder) is not the same as ${k}_{2}$ (external medium).

**Figure 5.**For the excitation of $\mathbf{TM}$ mode, a current generator ${I}_{n}^{TM}$ at the boundary of cylindrical nanowire, get distributed as internal (${I}_{int}^{TM}\left({k}_{1}a\right)$/${I}_{int}^{TM}\left({k}_{2}a\right)$) and external (${I}_{ext}^{TM}\left({k}_{1}a\right)$/${I}_{ext}^{TM}\left({k}_{2}a\right)$) current based on properties (${k}_{1}$/${k}_{2}$) of the nanowire, as shown in Figure 4a,b. Whereas ${Z}_{int}^{TM}\left({k}_{1}a\right)$/${Z}_{int}^{TM}\left({k}_{2}a\right)$ represents internal impedance of the nanocylinder with property ${k}_{1}$/${k}_{2}$ and ${Z}_{ext}^{TM}\left({k}_{2}a\right)$ is the external impedance of the nanowire.

**Figure 6.**For the excitation of $TE$ mode, a voltage generator ${V}_{n}^{TE}$ at the boundary of metallic nanowire, get distributed as internal (${V}_{int}^{TE}\left({k}_{1}a\right)$/${V}_{int}^{TE}\left({k}_{2}a\right)$) and external (${V}_{ext}^{TE}\left({k}_{1}a\right)$/${V}_{ext}^{TE}\left({k}_{2}a\right)$) voltage based on properties (${k}_{1}$/${k}_{2}$) of the internal medium as shown in Figure 4a,b. Whereas ${Y}_{int}^{n}\left({k}_{1}a\right)$/${Y}_{int}^{n}\left({k}_{2}a\right)$ represents internal admittance of the nanocylinder with property ${k}_{1}$/${k}_{2}$ and ${Y}_{ext}^{TM}\left({k}_{2}a\right)$ is the external admittance of the nanocylinder.

**Figure 7.**

**TE**mode: Comparison of ${Q}_{ext}$, ${Q}_{sca}$ and ${Q}_{abs}$ simulation for a 40 nm radius cylindrical nanowire using our proposed impedance model with the full-wave field solver. The values are normalized to unity for the extinction cross-sectional area efficiencies, with scattering and absorption plot scaled accordingly. The simulation result show negligible difference between the two models.

**Figure 8.**

**TM**mode: Comparison of ${Q}_{ext}$, ${Q}_{sca}$ and ${Q}_{abs}$ simulation for a 40 nm radius cylindrical nanowire using our proposed impedance model with the full-wave field solver. The values are normalized to unity for the extinction cross-section efficiency, with scattering and absorption plot scaled accordingly. The simulation result show negligible difference between the two models.

**Figure 9.**${\mathbf{TE}}_{n}$ mode extinction, scattering, and absorption cross-sectional areas efficiencies of cylindrical nanowire plotted with change in radius. The proposed model matches well with the full-wave field solution.

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**MDPI and ACS Style**

Alam, M.; Mahmood, A.; Azam, S.; Butt, M.S.; Haq, A.U.; Massoud, Y.
Impedance Model of Cylindrical Nanowires for Metamaterial Applications. *Nanomaterials* **2019**, *9*, 1104.
https://doi.org/10.3390/nano9081104

**AMA Style**

Alam M, Mahmood A, Azam S, Butt MS, Haq AU, Massoud Y.
Impedance Model of Cylindrical Nanowires for Metamaterial Applications. *Nanomaterials*. 2019; 9(8):1104.
https://doi.org/10.3390/nano9081104

**Chicago/Turabian Style**

Alam, Mehboob, Ali Mahmood, Shahida Azam, Madiha Saher Butt, Anwar Ul Haq, and Yehia Massoud.
2019. "Impedance Model of Cylindrical Nanowires for Metamaterial Applications" *Nanomaterials* 9, no. 8: 1104.
https://doi.org/10.3390/nano9081104