# Modeling semiconducting silicene nanostrips: electronics and THz plasmons

^{1}Department of Chemistry, Universidad Técnica Particular de Loja, Loja, Ecuador^{2}UNICARIBE Research Center, University of Calabria, Rende, Italy^{3}Facultad de Ciencias, Escuela Superior Politécnica de Chimborazo (ESPOCH), Riobamba, Ecuador^{4}Carrera de Matemática, Facultad de Ciencias, Escuela Superior Politécnica de Chimborazo (ESPOCH), Riobamba, Ecuador^{5}INFN-Laboratori Nazionali di Frascati, Frascati, Italy

Silicene nanostrips (SiNSs) have garnered significant attention due to their remarkable physical properties, making them an ideal candidate for numerous electronics and plasmonics applications. Their compatibility with current semiconductor technology further enhances their potential. This study aims to investigate the electronic and plasmonic properties of SiNSs with a minimum width of 100 nm using a semi-analytical model that utilizes the carrier velocity of silicene. The carrier velocity was calculated using density functional computations and refined through the GW approximation. Our results reveal that SiNSs with widths ranging from 100 to 500 nm exhibit small bandgaps within the range of a few meV, specifically ranging from 30 to 6 meV, respectively. Furthermore, all the nanostrips analyzed in this study exhibit a

## 1 Introduction

Silicene is an intriguing two-dimensional (2D) allotrope of silicon that shares a similar hexagonal lattice structure to graphene [1]. Composed of a single layer of silicon atoms, silicene is incredibly thin and flexible with a high surface area to volume ratio, giving it a wide range of potential applications [2]. One of the most promising aspects of silicene is its potential use in current semiconductor technology due to its silicon composition [3]. This opens up exciting possibilities for the development of advanced electronic devices that can take advantage of the unique electronic and optical properties of silicene [4]. For instance, its tunable bandgap can be controlled by applying an external electric field [5], making it suitable for electronic devices such as transistors. Additionally, silicene has a high thermal conductivity [6], which allows for efficient heat transfer and makes it useful in thermal management applications. Furthermore, the presence of unsaturated silicon atoms on its surface makes silicene highly reactive and capable of being functionalized with different chemical groups [7], making it useful for applications such as sensing and catalysis.

Silicene nanostrips (SiNSs), which typically have widths on the order of several tens of nanometers (i.e., silicene nanoribbons of

Despite numerous attempts to synthesize SiNSs using techniques such as scanning tunneling microscopy (STM) lithography [14], chemical vapor deposition (CVD) [15], chemical etching [16], and laser cutting [17]; there is a dearth of modeling approaches available for the study of very large nanostrips. While density functional theory (DFT) [18], tight-binding models [19], and Green’s function methods [20] are commonly used, these numerical methods can be challenging to implement for wide SiNSs due to the large number of atoms involved. As an alternative, semi-analytical models [21] can provide a useful tool for investigating the behavior of the material at a substantially lower computational cost. Such models can shed light on the electronic and plasmonic properties of SiNSs and guide experimental research in this field. By complementing experimental research, these models can help accelerate the development of silicene-based nanoelectronics and optoelectronics.

The present work aims to fill the gap in knowledge by investigating the electronic and plasmonic properties of wide SiNSs. The significance of studying these properties lies in understanding the behavior of SiNSs and how they can be used in practical electronic and photonic devices. Hence, a semi-analytical approach [22] is used in this work, which involves an *ab initio* many-body GW calculation to determine the charge-carrier velocity of freestanding silicene. This result is then integrated into the semi-analytical method to analyze the band structure, density of states (DOS), bandgap, and plasmon-frequency dispersion of SiNSs with a width of at least 100 nm. The study takes into account various factors such as excitation angle, effective electron mass, electron relaxation, and charge-carrier density to examine the plasmon dispersion and its tunability. This modeling technique can be extended to investigate the electronic and plasmonic properties of related systems such as nanostrips based on graphene and germanene.

## 2 Theoretical approach

### 2.1 Density functional computations

The ground-state properties of silicene are calculated using standard density functional theory (DFT) computations implemented in the Abinit software [23], specifically within the local density approximation (LDA) [24]. The Kohn-Sham (KS) electron wave functions are expanded in the plane-wave (

where *z* direction with a vacuum distance of 20

To calculate the electron band structure of silicene, we prepared two datasets of parameters. The first set involves a

### 2.2 GW calculations

To obtain an accurate calculation of the band dispersion of freestanding silicene, it is essential to incorporate many-body effects, which can be achieved by using the many-body GW self-energy method. As well-known, the GW method is a widely used approach for improving the accuracy of DFT calculations. The self-energy in the GW approximation is given by the expression [28]:

where

where

### 2.3 Semi-analytical model

To explore the plasmon characteristics of SiNSs that have a width of 100 nm or greater, we adopt the approach introduced by [21]. The investigation involves determining the plasmon frequency (

the parameters of Eq. 4 are detailed as follows.

•

•

•

•

•

•

•

The charge density (

where

From the experimental results presented by [30], it has been demonstrated that in graphene nanostrips with widths ranging from 155 to 480 nm, there exist two distinct resonance modes: the surface plasmon and the edge plasmon. It is also expected that these modes are present in SiNSs. In particular, the edge plasmon mode can be selectively tuned by altering the ribbon width [31], while the surface plasmon mode is found to be more sensitive to various external factors, including doping levels, temperature, electron mobility, and the angle of plasmon excitation [32]. These findings have significant implications for the design of nanoscale electronic and photonic devices. Indeed, by controlling the dimensions of the nanostrips and selectively tuning the plasmon modes, it may be possible to engineer novel devices with tailored optical and electronic properties.

Eq. 4 is a mathematical expression that describes only the dispersion of the surface plasmon mode. While this equation does not account for the nature of the edge plasmon mode, it is still an efficient means of calculating the frequency dispersion of surface plasmons. Furthermore, this expression is consistent with experimental observations, which have shown that the plasmon wavelength follows the sample length, with the sample length being much larger than both the vacuum distance and the ribbon width [33]. For simplicity, we use the term “plasmon” exclusively to refer to the surface plasmon mode in this work.

To further customize the analysis of plasmonic properties in a specific context, Eq. 4 can be modified as needed, for instance, using.

(i) The Fermi level (

(ii) The intrinsic semiconductor behavior [35] as:

In Eq. 7, it is assumed that

Due to Eq. 4 being a straightforward analytical expression, when the plasmon damping (

From the physical point of view, this effect can be caused by various mechanisms, including scattering, absorption, and radiation, which lead to a loss of plasmon energy. As a result, the plasmon response can move towards larger momenta, which corresponds to a higher frequency or shorter wavelength.

Moreover, the notion of a complex dielectric function (

thus,

With this in mind, it is now feasible to compute the effective electron mass in Eq. 4 by incorporating the charge-carrier velocity (

and the bandgap (

here,

As noted, the charge-carrier velocity of silicene is the critical parameter in Eq. 4 and is the foundation of the semi-analytical model. Then, to estimate the charge-carrier velocity accurately, we employed either GW or LDA-DFT calculations to perform a linear fit of the

here,

On the other hand, to estimate the band structure of SiNSs, it is crucial to take into account the quasi-one-dimensional confinement of charge carriers. This confinement leads to the formation of multiple energy sub-bands (

here the integer number

### 2.4 Lorentz function

Now, to show the plasmon spectrum (i.e., the maximum of the plasmon peak) for selected values of

here,

The Lorentzian line function is a widely used model for describing spectroscopic features in physical systems like ions, atoms, and molecules, as well as in SiNSs. This study employs frequency units for

## 3 Results and discussions

### 3.1 Freestanding silicene

To begin, we analyze the band structure of freestanding silicene using a

**FIGURE 1**. Band structure of freestanding silicene along the high symmetry

It is worth noting that the LDA results show a lowering of the bands, while the gaps at the Γ (3.3

We now focus on the region around the Dirac cone (K point) by examining energy-momentum data obtained from GW calculations of freestanding silicene. We have considered the first conduction (

**FIGURE 2**. The low-energy band structure and charge-carrier velocity of freestanding silicene near the K point and around the Fermi level. **(A)** Displays the energy bands of silicene with the **(B)** The color plot of the charge-carrier velocity (10^{6} m s^{−1}) of silicene as a function of band dispersion vs. wave vector. The negative values refer to the velocity in the valence band.

Figure 2B displays the linear behavior of charge-carrier velocity, although only within a narrower energy range (

Specifically, Table 1 compares the estimated values by LDA-DFT (^{−1}) and GW (^{−1}) calculations. The LDA-DFT calculation underestimated the GW value by 27.22%. Currently, there is no consensus on the experimental charge-carrier velocity of silicene. To gain a deeper understanding of our findings, we compared the results obtained for silicene with those of graphene at the same theoretical levels (LDA-DFT-graphene (Supplementary Figure S2) and GW-graphene (Supplementary Figure S3). Interestingly, we found that the charge-carrier velocity of silicene is 33.63% lower than that of graphene. It is worth noting that the GW calculation for graphene showed only a slight overestimation of 1.64%, which closely matched the experimental result for graphene (Exp-graphene) [42]. Similarly, as in silicene, LDA-DFT calculations for graphene yield a charge-carrier velocity that is 24.64% lower than the experimentally measured value. This discrepancy highlights the limitations of the LDA-DFT method and the need to employ more advanced calculations. Additionally, these results confirm that the GW approximation is a reliable method for predicting the charge-carrier velocity of silicene, and future experiments are likely to produce similar outcomes.

**TABLE 1**. The charge-carrier velocity of silicene is estimated by LDA and GW calculations. The results are compared with the experimental (Exp-graphene) [42] y theoretical (GW-graphene) values of graphene.

### 3.2 Electronic properties of SiNSs

Equation 11 demonstrates that there is an inverse relationship between the bandgap and ribbon width. Specifically, as the width of the ribbon increases, the bandgap value decreases exponentially. Therefore, if the ribbon width were to approach infinity (

**FIGURE 3**. Comparison of the estimated bandgap values for SiNSs with widths ranging from **(A)** 10–50 nm and **(B)** 100–500 nm, using both GW (^{−1}, red line) and DFT-LDA (^{−1}, blue line) calculations.

The impact of ribbon width on the bandgap is most pronounced apparently in narrow SiNSs with widths ranging from 10 to 50 nm, as shown in Figure 3A, where the bandgap decreases from 0.3 eV to 50 meV. Conversely, wider SiNSs with widths ranging from 100 to 500 nm, shown in Figure 2B, exhibit a decrease in bandgap from 30 meV to 5 meV. In both cases, the bandgap decreases by a factor of six. Moving forward, we will focus on the electronic properties of SiNSs using the charge-carrier velocity of silicene obtained through GW calculations (i.e., ^{-1}).

Table 2 presents information on the bandgaps and effective electron masses for various nanostrips with different widths ranging from 100 to 500 nm. The data reveals a noteworthy trend: the wider the nanostrip, the smaller the bandgap. This trend is evident in the 100 nm wide nanostrip, which has a bandgap of approximately 30 meV, compared to the 500 nm wide nanostrip, which has a bandgap of approximately 6 meV. Furthermore, when comparing the same nanostrips, the effective electron masses show the same trend. The effective electron mass values obtained for the different nanostrips are consistent with both experimental findings and predictions for graphene nanostrips [43], [45], in terms of orders of magnitude. These results suggest that the properties of the SiNSs are similar to those of graphene nanostrips.

**TABLE 2**. Bandgap (meV) and effective mass (

In Figure 4, we now present the band structure (Figures 4A, C) and density of states (DOS) (Figures 4B, D) of the 100 nm and 500 nm wide nanostrips, in a

**FIGURE 4**. Band structure and density of states (DOS) of two SiNSs with different widths: **(A, B)** 100 nm and **(C, D)** 500 nm. The DOS is calculated from the energy-momentum data list using a conventional histogram with equal bin widths, and the red line represents the smoothed curve of the histogram.

Upon comparison, it becomes apparent that there is a quadratic band dispersion of the conduction and valence bands near the *ab initio* methods such as DFT, the main issue is that the systems analyzed in this study are too broad for current computers to handle. Then, the semi-analytical model proves advantageous in this regard and produces reasonable outcomes. In light of this, we can observe that in the same energy range, the 500 nm wide nanostrip (Figure 4C) displays a considerably greater number of bands in comparison to the 100 nm wide nanostrip (Figure 4D). It is also worth noting that with an increase in the width of nanostrips, the energy gap between the bands decreases, and they tend to converge, whenever possible, to the same energy state, similar to what we observe in silicene with the corresponding

Figures 4B, D show the DOS of the 100 nm and 500 nm wide nanostrips, respectively. These figures display several peaks corresponding to the bands in the band structure plot. Interestingly, smoothing the DOS histogram (red line) can lead to a narrower bandgap of the system. For example, the bandgap of the 100 nm wide nanostrip is roughly 23 meV, while that of the 500 nm wide nanostrip is around 4 meV. This results in a 23% and 33% reduction in the bandgap, respectively (Table 2).

### 3.3 Charge density effect

In this section, we investigate the plasmonic properties of SiNSs, focusing on the effect of charge carrier density (

Indeed, previous studies have reported that the charge carrier density in isolated graphene nanostrips is approximately ^{−1} [39],[46]. Similar results are expected for SiNSs. However, this value can vary depending on factors such as doping, ribbon size, or geometric factors. Then, Eq. 5 can be used as a starting point to adjust the charge density to the desired value (see Table 3). Interestingly, as the distance between adjacent nanostrips increases, the charge density decreases. For instance, when the vacuum distance between nanostrips increases from 10 nm to 100 nm, the charge density reduces by one order of magnitude. However, it is worth noting that, in this case, the Fermi level (

**TABLE 3**. 2D charge density and Fermi level shift, as influenced by different vacuum distances between adjacent nanostrips. The expression used to modulate the charge density is

Figure 5A shows the dispersion of plasmon frequency-momentum for a 100 nm wide nanostrip as a function of the reciprocal wave vector (

**FIGURE 5**. **(A)** Plasmon frequency dispersion by altering the vacuum distance between neighboring nanostrips. **(B)** The maximum of the plasmon peak for specific momentum values. **(C)** Density plot of the plasmon frequency-momentum dispersion as a function of density concentration and momentum. Results for a 100 nm wide nanostrip with:

Additionally, Table 3 indicates that a separation distance of 10 nm results in a high charge density (^{−2}) and a shift in the Fermi level of 0.38 eV, which may lead to excessive doping for a 100 nm wide nanostrip with a small bandgap of a few meV (∼30 meV). Therefore, we focus on a charge density of ^{−2}, which is a commonly used value for modeling graphene or similar materials. This choice results in a Fermi level shift of only 0.12 eV and a separation distance of 100 nm is adequate. Furthermore, this vacuum distance ensures that the system can be treated as a quasi-isolated SiNS.

Considering this, Figure 5B shows the dispersion of the maximum of the plasmon peak for a 100 nm wide nanostrip with a charge density of ^{−2} at selected momentum (^{−2}. The analysis reveals two significant observations: i) forbidden regions for plasmons at values of

### 3.4 Effective mass effect

The effective electron mass (

Eq. 10 contains this parameter, which is only affected by changes in the bandgap, such as those resulting from modifications to the ribbon width (Table 2) since the charge-carrier velocity (^{−1}) remains constant in this work. With this in mind, Figure 6A shows the plasmon dispersion of SiNSs with widths ranging from 100 to 500 nm as a function of the momentum (

**FIGURE 6**. **(A)** Plasmon frequency dispersion by altering the effective electron mass which is modulated by increasing the ribbon width from 100 to 500 nm. **(B)** The maximum of the plasmon peak at ^{−1}. **(C)** Density plot of the plasmon frequency-momentum dispersion as a function of density concentration and momentum. The parameters are fixed as: ^{−2},

Figure 6B provides compelling evidence of the controllability and tunability of plasmon response, showing that increasing the width of the nanostrip or decreasing the effective electron mass results in an increase in plasmon frequency. This figure focuses on a frequency range of 0.1–3 THz, which is highly relevant for plasmonic applications in silicene, as mentioned earlier. Therefore, nanostrip systems ranging from 100 to 500 nm hold promise as potential candidates for such applications. Furthermore, Figure 6C demonstrates that there are no forbidden regions for plasmons and confirms that a decrease in the effective electron mass leads to an increase in plasmon frequency by approximately 25 THz (red region).

A final remark, Fei et al. [30] have successfully prepared graphene nanostrips with widths similar to those examined in this study (i.e.,

### 3.5 Excitation angle effect

The plasmon excitation angle (

**FIGURE 7**. **(A)** Plasmon frequency dispersion by altering the plasmon excitation angle from **(B)** The maximum of the plasmon peak at ^{−1}. **(C)** Density plot of the plasmon frequency-momentum dispersion as a function of density concentration and momentum. Results for a 100 nm wide nanostrip with: ^{−2},

Figure 7C exhibits the existence of forbidden regions where plasmons are prohibited, and as the angle increases. Interestingly, these prohibited regions can also be observed by increasing the momentum value (as evidenced by the extended purple region). Moreover, it is important to highlight that the highest spectral intensity is achieved with small angles (

### 3.6 Electron relaxation effect

Electron relaxation rate (

Then, we focus on the electron relaxation rate in nanostrip of 100 and 500 nm wide. Additionally, this parameter is crucial in providing the values for electronic mobility and plasmon relaxation rate (Eq. 8) as shown in Table 4. Particularly, the electron mobility (

**TABLE 4**. The calculated values of electron relaxation rate (

Figure 8A depicts the plasmon frequency-momentum dispersion of a 100 nm wide nanostrip as a function of the increasing electron relaxation rate. Notably, the curves appear to overlap, prompting us to investigate whether this effect is attributable to the width of the strip. To that end, we analyzed a 500 nm wide nanostrip (as shown in Figure 8B) but found that the plasmon frequency-momentum dispersion followed the same pattern.

**FIGURE 8**. Plasmon frequency dispersion by altering the electron relaxation rate: Results for **(A)** 100 nm wide nanostrip and **(B)** 500 nm wide nanostrip (^{−2},

Given that this effect is not due to strip width, we continue the analysis to a lower momentum regime for a 100 nm wide nanostrip, as depicted in Figure 9. We observe in Figure 9A that the plasmon frequency continues to be proportional to the square root of

(i) The plasmonic response moves towards higher momentum values.

(ii) The frequency of the plasmonic response drops significantly.

**FIGURE 9**. **(A)** Plasmon frequency dispersion by altering the electron relaxation rate. **(B)** The maximum of the plasmon peak at ^{-1}. **(C)** Density plot of the plasmon frequency-momentum dispersion as a function of density concentration and momentum. Results for a 100 nm wide nanostrip with: ^{−2},

Apart from the significant influence of the electron relaxation rate, Figure 9B shows that even a 100 nm wide SiNS subjected to the electron relaxation rate effect displays plasmon responses in the frequency range of 0.1–3 THz, suggesting even greater adjustable plasmonic properties. Figure 9C demonstrates the existence of forbidden regions for plasmons, which become more prominent as the electron relaxation rate and momentum increase (purple region). As previously mentioned, increasing the electron relaxation rate results in a significant reduction in the plasmon frequency. For instance, in a 100 nm wide nanostrip, the plasmon frequency reaches a maximum of around 2 THz (red region).

Hence, Eq. 4 provides a comprehensive understanding of the parameters that influence the plasmon response, including its frequency and dispersion. By thoroughly examining these parameters, one can manipulate them individually or in combination to customize the plasmon response for a specific application.

## 4 Conclusion

In summary, we have utilized a semi-analytical model that relies on the charge-carrier velocity of silicene as its main input parameter. To determine the charge-carrier velocity, we conducted LDA-DFT computations, resulting in ^{−1}, which was further refined using the GW approximation, resulting in ^{−1}. We then incorporated this GW value into the proposed semi-analytical modeling approach, which encompassed a wide range of parameters and variables to comprehensively investigate the behavior and characteristics of SiNSs with widths of 100 nm or greater.

Our results indicate that SiNSs with widths ranging from 100 nm to 500 nm possess bandgap values in the range of a few meV. Specifically, we found that nanostrips with widths of 100, 200, 300, 400, and 500 nm exhibit bandgaps of approximately 30, 15, 10, 8, and 6 meV, respectively. Furthermore, the effective electron mass for these systems ranges from

Furthermore, our study provides a comprehensive investigation into the plasmonic properties of various nanostrips, including the effects of ribbon width, charge density, plasmon excitation angle, and electron relaxation rate. Particularly, we demonstrate that by adjusting these parameters individually or in combination, it is possible to achieve the desired plasmon resonance, enabling precise control and customization of plasmonic properties to meet specific application requirements. As the main findings, all the nanostrips examined in this study display a

Our study provides a thorough understanding of the electronic and plasmonic properties of SiNSs, which is essential for the advancement of future nanodevices. The insights gained from our research can serve as a valuable reference for future experiments aimed at verifying and building upon our findings.

## Data availability statement

The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.

## Author contributions

Conceptualization, TT and CVG; Methodology, MB, MJMS, and MLPR; Validation, TT, MG, and CVG; Formal Analysis, CG; Resources, TT; Data Curation, TT; Writing—Original Draft Preparation, CVG; Writing—Review and Editing, CVG. TT; Funding Acquisition, TT. All authors have read and agreed to the published version of the manuscript. All authors contributed to the article and approved the submitted version.

## Funding

This work was funded by Universidad Técnica Particular de Loja (UTPL-Ecuador) under the project: “Análisis de las propiedades térmicas del grafeno y zeolita” Grand No: PROY_INV_QU_2022_362.

## Acknowledgments

TT, MG, and CG wish to thank the Ecuadorian National Department of Sciences and Technology (SENESCYT). This work was partially supported by LNF-INFN: Progetto HPSWFOOD Regione Lazio—CUP I35F20000400005.

## Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

## Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2023.1198214/full#supplementary-material

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Keywords: silicene, silicene nanostrips, THz plasmons, electronic properties, group velocity

Citation: Tene T, Guevara M, Borja M, Mendoza Salazar MJ, Palacios Robalino MdL, Vacacela Gomez C and Bellucci S (2023) Modeling semiconducting silicene nanostrips: electronics and THz plasmons. *Front. Phys.* 11:1198214. doi: 10.3389/fphy.2023.1198214

Received: 31 March 2023; Accepted: 27 April 2023;

Published: 10 May 2023.

Edited by:

Li Li, Harbin Institute of Technology, ChinaReviewed by:

Mauludi Ariesto Pamungkas, University of Brawijaya, IndonesiaShengxuan Xia, Hunan University, China

Copyright © 2023 Tene, Guevara, Borja, Mendoza Salazar, Palacios Robalino, Vacacela Gomez and Bellucci. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Cristian Vacacela Gomez, vacacela@lnf.infn.it