# Resolving the Mechanism of Acoustic Plasmon Instability in Graphene Doped by Alkali Metals

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## Abstract

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## 1. Introduction

## 2. Theoretical Formulation

#### 2.1. Calculation of the Surface Electronic Excitations Spectra

#### 2.2. Calculation of the 2D Dynamical Polarizability Function $\alpha \left(\omega \right)$

#### 2.3. Calculation of the Substrate Dielectric Function

#### 2.4. Reduced Model

#### 2.5. Computational Details

## 3. Results and Discussion

#### Resolving the Mechanism of the AP Instability

## 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.**Schematic representation of the AC${}_{8}$ crystal deposited on a dielectric substrate described by a local dielectric function ${\u03f5}_{S}\left(\omega \right)$. The substrate (blue) occupies the region $z<-h$, the graphene layer (yellow) is in the $z=0$ plane and the alkali atom (turquoise) layer is in the $z=-d$ plane.

**Figure 2.**The interband contribution to the dynamical polarizability ${\alpha}^{\mathrm{inter}}\left(\omega \right)$ for the KC${}_{8}$ (black), CsC${}_{8}$ (orange) and doped graphene (red dashed). The graphene is doped so that the Fermi energy is 1eV above Dirac point, which corresponds to the doping of the $\pi $ bands in the KC${}_{8}$ and CsC${}_{8}$.

**Figure 3.**Schematic representation of the reduced model. The alkali atom layer is approximated by ‘massive’ 2D electron gas (parabolic $\sigma $ band), and the graphene layer is described by the ‘massless’ Dirac fermion (MDF) approximation (conical $\pi $ band).

**Figure 4.**The electronic band structure in (

**a**) KC${}_{8}$ and in (

**b**) CsC${}_{8}$. The magenta line denotes the lowest unoccupied band (LUCB). The blue circles represent the parabolic fit $E\left(K\right)={E}_{F\sigma}+\frac{{\hslash}^{2}{K}^{2}}{2{m}_{\sigma}}$ of the alkali atom $\sigma $ band.

**Figure 5.**The ab initio spectra of the electronic excitations $S(\mathbf{Q},\omega )$ in the (

**a**) KC${}_{8}$ and (

**b**) CsC${}_{8}$ deposited on dielectric Al${}_{2}$O${}_{3}$ surface with $h=5.92$ Å and $h=6.13$ Å respectively (i.e., the separation between the dielectric surface and the alkali atom layer is chosen to be 3 Å). The ab initio spectra of the electronic excitations in the self-standing (

**c**) KC${}_{8}$ and (

**d**) CsC${}_{8}$. The magenta dots in (

**a**,

**b**) denote the DP and AP dispersion relations in the self-standing samples, i.e., the DP and AP in (

**c**,

**d**). The energy loss function $-\Im \{1/\u03f5(\mathbf{Q},\omega \left)\right\}$ in the self-standing (

**e**) KC${}_{8}$ and (

**f**) CsC${}_{8}$ was obtained using the reduced model.

**Figure 6.**AP peak for $Q=0.1$a.u. when various numbers of the unoccupied bands are omitted from the calculation (n) for the (

**a**) CsC${}_{8}$ and (

**b**) KC${}_{8}$. (

**c**) Intensity of the AP peak as a function of n for the CsC${}_{8}$ (black line) and KC${}_{8}$ (red line).

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**MDPI and ACS Style**

Marušić, L.; Kalinić, A.; Radović, I.; Jakovac, J.; Mišković, Z.L.; Despoja, V.
Resolving the Mechanism of Acoustic Plasmon Instability in Graphene Doped by Alkali Metals. *Int. J. Mol. Sci.* **2022**, *23*, 4770.
https://doi.org/10.3390/ijms23094770

**AMA Style**

Marušić L, Kalinić A, Radović I, Jakovac J, Mišković ZL, Despoja V.
Resolving the Mechanism of Acoustic Plasmon Instability in Graphene Doped by Alkali Metals. *International Journal of Molecular Sciences*. 2022; 23(9):4770.
https://doi.org/10.3390/ijms23094770

**Chicago/Turabian Style**

Marušić, Leonardo, Ana Kalinić, Ivan Radović, Josip Jakovac, Zoran L. Mišković, and Vito Despoja.
2022. "Resolving the Mechanism of Acoustic Plasmon Instability in Graphene Doped by Alkali Metals" *International Journal of Molecular Sciences* 23, no. 9: 4770.
https://doi.org/10.3390/ijms23094770