# Energy Relaxation and Electron–Phonon Coupling in Laser-Excited Metals

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## Abstract

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## 1. Introduction

## 2. Method

#### 2.1. Formalism

#### 2.2. Calculation Details

## 3. Results and Discussions

#### 3.1. Aluminum

#### 3.2. Copper

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Eliashberg function ${\alpha}^{2}F\left(\omega \right)$ and (

**b**) phonon density of states $F\left(\omega \right)$ of aluminum at different electron temperatures (shifted along the y-axis for better visibility). The dashed black line is the low temperature result of Brown et al. [48].

**Figure 2.**Results for aluminum. (

**a**) Electron density of states $g(\epsilon ,{T}_{e})$ with increasing electron temperature and Fermi distribution function $f(\epsilon ,\mu \left({T}_{e}\right),{T}_{e})$ (dashed lines) for two electron temperatures at 315 K and $50,000$ K; (

**b**) chemical potential $\mu \left({T}_{e}\right)$ and (

**c**) electron density of states at chemical potential $g\left[\mu \left({T}_{e}\right)\right]$ as a function of electron temperature; (

**d**) electron temperature-dependent electron–phonon coupling factor ${G}_{ep}\left({T}_{e}\right)$, compared with various theoretical calculations. The available theoretical data are predicted by Lin et al. [44], Müller et al. [45] (their estimations contain two different physical conditions), Petrov et al. [46], Waldecker et al. [33], Brown et al. [48], and Medvedev et al. [49].

**Figure 3.**Results for aluminium. Eliashberg function ${\alpha}^{2}F\left(\omega \right)$ of partial branches of (

**a**) TA1, (

**b**) TA2, and (

**c**) LA at different electron temperatures; (

**d**) second moment of Eliashberg function $\lambda \langle {w}^{2}\rangle $ and (

**e**) electron–phonon coupling constant $\lambda $ for total and three different branches (TA1, TA2, LA) with increasing electron temperature; (

**f**) electron temperature-dependent electron–phonon coupling factor ${G}_{ep}\left({T}_{e}\right)$ for three partial branches (TA1, TA2, LA).

**Figure 4.**(

**a**) Eliashberg function ${\alpha}^{2}F\left(\omega \right)$ and (

**b**) phonon density of states $F\left(\omega \right)$ of copper at different electron temperatures (shifted along the y-axis for better visibility). The black dashed line is the result for the phonon-DOS by Ono [74].

**Figure 5.**Results for copper. (

**a**) Electron density of states $g(\epsilon ,{T}_{e})$ with increasing electron temperature and Fermi distribution function $f(\epsilon ,\mu \left({T}_{e}\right),{T}_{e})$ (dashed lines) for two electron temperatures at 315 K and 50000 K; (

**b**) chemical potential $\mu \left({T}_{e}\right)$ and (

**c**) electron density of states at chemical potential $g\left[\mu \left({T}_{e}\right)\right]$ as a function of electron temperature; (

**d**) electron temperature-dependent electron–phonon coupling factor ${G}_{ep}\left({T}_{e}\right)$, compared with various theoretical calculations.The available theoretical data are estimated by Lin et al. [44], Migdal et al. [52], Ji et al. [51], Brown et al. [48], Smirnov [53], and Medvedev et al. [49]. The green crosses are experimentally extracted values by Cho et al. [75].

**Figure 6.**Results for copper. Eliashberg function ${\alpha}^{2}F\left(\omega \right)$ of partial branches of (

**a**) TA1, (

**b**) TA2 and (

**c**) LA at different electron temperatures; (

**d**) second moment of Eliashberg function $\lambda \langle {w}^{2}\rangle $ and (

**e**) electron–phonon coupling constant $\lambda $ for total and three different branches (TA1, TA2, LA) with increasing electron temperature; (f) electron temperature-dependent electron–phonon coupling factor ${G}_{ep}\left({T}_{e}\right)$ for three partial branches (TA1, TA2, LA).

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**MDPI and ACS Style**

Zhang, J.; Qin, R.; Zhu, W.; Vorberger, J. Energy Relaxation and Electron–Phonon Coupling in Laser-Excited Metals. *Materials* **2022**, *15*, 1902.
https://doi.org/10.3390/ma15051902

**AMA Style**

Zhang J, Qin R, Zhu W, Vorberger J. Energy Relaxation and Electron–Phonon Coupling in Laser-Excited Metals. *Materials*. 2022; 15(5):1902.
https://doi.org/10.3390/ma15051902

**Chicago/Turabian Style**

Zhang, Jia, Rui Qin, Wenjun Zhu, and Jan Vorberger. 2022. "Energy Relaxation and Electron–Phonon Coupling in Laser-Excited Metals" *Materials* 15, no. 5: 1902.
https://doi.org/10.3390/ma15051902