# A Theory of Dynamical Responses for Metal Films: Surface Roughness Effects

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## Abstract

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## 1. Introduction

## 2. Analytical Expressions for the Dynamical Response Function

#### 2.1. Generic Formalism

#### 2.2. Macroscopic Limit

#### 2.3. Kinetic Theory

## 3. Application to Plasma Waves

## 4. Summary

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Numerically computed ${\mathcal{H}}_{l,{l}^{\prime}}^{+}$ (

**a**–

**c**) and ${\mathcal{H}}_{l,{l}^{\prime}}^{-}$ (

**d**–

**f**), both normalized by ${\omega}_{p}^{2}$. (

**a**,

**d**): real parts; (

**b**,

**e**): imaginary parts; (

**c**,

**f**): diagonal elements. Parameters: $\omega /{\omega}_{p}=0.7$, $L{\omega}_{p}/{v}_{F}=100$, $\tau {\omega}_{p}=50$, $k{v}_{F}/{\omega}_{p}=0.0001$ and $p=0$.

**Figure 2.**Numerically computed ${G}_{l}^{+}$ (

**a**) and ${G}_{l}^{-}$(

**b**). Parameters are the same as for Figure 1.

**Figure 3.**Maps of $log\left(\left|\mathrm{Det}\left(\frac{1}{{\overline{\omega}}^{2}\mathbb{I}-{\mathcal{H}}^{\pm}}\right)\right|\right)$ for $L{\omega}_{p}/{v}_{F}=100$ (

**a**–

**d**) and $L{\omega}_{p}/{v}_{F}=500$ (

**e**–

**h**). $\tau {\omega}_{p}=50$.

**Figure 4.**Maps of the quantity $|\frac{1}{{\u03f5}^{+}(k,\omega )}{|}^{2}+{\left|\frac{1}{{\u03f5}^{-}(k,\omega )}\right|}^{2}$ showing the dispersion of SPWs hosted on a metal film of width L and surface roughness p. (

**a**–

**d**) are for $L{\omega}_{p}/{v}_{F}=100$ and (

**e**–

**h**) are for $L{\omega}_{p}/{v}_{F}=500$. $\tau {\omega}_{p}=50$.

**Figure 5.**SPW peaks for various combinations of k and p. (

**a**–

**h**): $L{\omega}_{p}/{v}_{F}=100$; (

**i**–

**p**): $L{\omega}_{p}/{v}_{F}=500$. (

**a**–

**d**,

**i**–

**l**): symmetric modes; (

**e**–

**h**,

**m**–

**p**): anti-symmetric modes. $\tau {\omega}_{p}=50$.

**Figure 6.**FWHM of SPW resonance peaks at various values of p. (

**a**–

**e**,

**k**–

**o**): symmetric SPWs; (

**f**–

**j**,

**p**–

**t**): anti-symmetric SPWs. $\tau {\omega}_{p}=50$.

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**MDPI and ACS Style**

Praill, S.; Lawton, C.; Balable, H.; Deng, H.-Y.
A Theory of Dynamical Responses for Metal Films: Surface Roughness Effects. *Solids* **2023**, *4*, 268-286.
https://doi.org/10.3390/solids4030017

**AMA Style**

Praill S, Lawton C, Balable H, Deng H-Y.
A Theory of Dynamical Responses for Metal Films: Surface Roughness Effects. *Solids*. 2023; 4(3):268-286.
https://doi.org/10.3390/solids4030017

**Chicago/Turabian Style**

Praill, Sam, Charlotte Lawton, Hasan Balable, and Hai-Yao Deng.
2023. "A Theory of Dynamical Responses for Metal Films: Surface Roughness Effects" *Solids* 4, no. 3: 268-286.
https://doi.org/10.3390/solids4030017