Correlation between energy band transition and optical absorption spectrum in bilayer armchair graphene nanoribbons

First of all, we considered semi-conducting groups $3p$ and $3p + 1$ of armchair-edge structures. The spectra functions of group $3p$ in BL-AGNRs for different ribbon widths correspond to $M\, = \,3,\,\,6,\,\,9$ as represented in figure 2.

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In figure 2(a), the distinction of the structure peaks in the range $\omega = \left[ {0,\,4{t_0}} \right]$ indicates a remarkable variation of the position, the height, and the number of optical peaks with the change of the ribbon width $M.$ We noticed that the appearance of the prominent peak near $2{t_0}$ stems from the concave-upward parabolic $\pi$ bands at ${E^{\,v}}\sim - {t_0}$ (in valence bands) and the concave-downward parabolic ${\pi ^ * }$ bands at ${E^c}\sim{t_0}$ (in conduction ones) [22]. More importantly, the previous studies pointed out that the complete flat bands at ${E^{c,v}} = \pm {t_0}$ occur in the energy bands of the odd dimer lines $M,$ where the electrons are localized the most significantly [20, 44]. This can be considered as the reason for the existence of the main peak at $2{t_0}$ in the absorption spectra for $M = 3$ and $9,$ and it is more prominent than that for $M = 6.$ Moreover, the optical spectroscopy of SL-AGNRs (with $M = 3$) also exists a striking peak at this crucial frequency position, as shown in the inset on the upper right corner of figure 2(a).

When comparing the optical spectra in BL-AGNRs and SL-AGNRs at the same ribbon width $\left( {M = 3} \right),$ it can be seen that the main differences are the positions of peaks ${\omega _1}s$ in the low-frequency region of the former structure closer to the zero-point than those of the latter structure, i.e. ${\omega _1} = 0.7624{t_0}$ in BL-AGNRs and ${\omega _1} = 0.8317{t_0}$ in SL-AGNRs. In addition, the number of absorption peaks in BL-AGNRs is more outstanding than in SL-AGNRs. The physical explanation is that the energy structure in BL-AGNRs has a more remarkable number of subbands than in SL-AGNRs with the same value of $M$ [45]. This outcome suggests that when increase in the number of layers to three or more, due to the increment in the number of subbands and the decrease in the gap size [46], it likely causes a significant variation in the absorption spectrum, i.e. a rise in the number of subbands will create new excitation channels, and therefore an increase in the number of peaks. In addition, the narrowing of the electronic gap will result in the redshift as well as the decrease in the height of the threshold absorption peak.

In summary, when the number of these subbands increases, the excitation channels are notably diversified, causing a vigorous increment in the number of absorption peaks. This result is in agreement with the experimental observation on the optoelectronic devices in which it unveiled the dependence of optical properties on the ribbon width as well as the number of layers [47, 48].

It can be also observed that the subpeaks in the left- and right-hand neighborhoods of the main peak move in two opposite directions, which are a redshift (to lower frequencies) and a blueshift (to higher frequencies) respectively. At once, the heights of subpeaks also decline formidably when the width $M$ increases. The above analyses show that the peak structure in the low-frequency absorption spectra varies strongly with the width $W$ of ribbon structures.

It is worth noting from figure 2(a) that the height of the prominent peak increases with increasing the ribbon width $M.$ Our further check on different width structures also confirmed this interesting outcome. This result stems from a special feature in the electronic structures of group $3p$ in which some low-slope or complete flat bands exist near the crucial energy points $- {t_0}$ and ${t_0}.$ When the width increases, the number of these bands also raises, therefore the strength of the associate optical peak is also enhanced.

In figure 2(b), the spectral structure of the 1st peaks is observed even more obviously for both BL-AGNRs (solid lines) and SL-AGNRs (dashed lines) in the considered range $\left[ {0,0.9{t_0}} \right].$ The position of these peaks seems to displace significantly to the lower-frequency range, i.e. the threshold absorption frequencies of BL-AGNRs and SL-AGNRs are correlative to ${\omega _1}s = 0.7624{t_0}\,\left( {M = 3} \right),$ $0.4158{t_0}\,\left( {M = 6} \right),$ $0.2772{t_0}\,\left( {M = 9} \right),$ and ${\omega _1}s = 0.8317{t_0}$ $\left( {M = 3} \right),$ $0.5025{t_0}\,\left( {M = 6} \right),$ $0.3639{t_0}\,\left( {M = 9} \right),$ respectively. Concurrently, the spectral peak shape is adjusted remarkably when changing the ribbon width $M,$ especially it modifies the width of the 1st peak. It is worth noting that the peak width can be calculated by the full width at half maximum technique (FWHM) [49], as an example shown in the inset of figure 2(b) for a BL-AGNR of $M = 3.$ FWHM is defined as the difference between the two values of the independent variable ${x_1}$ and ${x_2},$ in which the dependent variable is half of the maximum amplitude $A\left( \omega \right) = {A_{\max }}\left( \omega \right)/2.$ Within this approach, the widths of the spectral peak in BL-AGNRs are ${\text{FWHM}} =$ $0.0204{t_0},\,\,0.0244{t_0},\;0.0249{t_0}$ corresponding to three ribbons of widths $M = 3,\,6$ and $9.$ It is noted that for group $3p,$ the smaller the ribbons are, the narrower the peak widths are, and vice versa. The phenomenon can be interpreted by the fact that for the large values of armchair lines $M,$ both the number of the transition channels forming the peaks ${\omega _1}s$ as well as the electronic distribution at these peaks are increasing, resulting in the broadening of FWHM.

To evidence a correlation between the inter-$\pi$-band excitations and the formation of optical peaks, we examined the energy bands and the DOSs of BL-AGNRs as shown in figure 2(c). It is worth noting that in this study, the pairs of the conduction and valence subbands are symbolized by the indices $J$ in an increasing sequence accounted from the Fermi level, with ${J^{cv}} = \pm 1,\, \pm 2,\,\ldots,\, \pm N,$ where the signs $\left( + \right)$ and $\left( - \right)$ represent the unoccupied $\left( {{E^c}} \right)$ and the occupied states $\left( {{E^{\,v}}} \right).$ For two groups of semiconducting and insulating ribbons, the previous studies indicated that the formation of the absorption peaks comes from the allowed transitions among the energy levels belonging the same index $J$ [21, 22]. For instance, the excitation to form the 1st peaks is from ${J^{\,v}} = - 1$ ($\pi$ band) to ${J^c} = 1$ (${\pi ^*}$ band), meaning that the minimum energy difference is equal to the gap size. More interestingly, because of the van der Waals interactions which are presented via the interlayer parameters (as mentioned in section 2.1), the energy dispersion of BL-AGNRs shows a discrepancy from that of SL-AGNRs. Although electron–hole symmetry can be observed with SL-AGNRs, the interlayer atomic interactions ${t_i}\,\,\left( {i = 1,\,3,\,4} \right)$ lead to the asymmetry between the energy levels around the Fermi energy ${E_{\text{F}}} = 0$ in BL-AGNRs [22, 50]. It results in the fact that the positions of the 1st peaks in SL-AGNRs and BL-AGNRs are defined slightly differently, i.e. ${\omega _1} = 2E_{J = 1}^c = 2\left| {E_{J = - 1}^v} \right|$ and ${\omega _1} =$ $E_{J = 1}^c + \left| {E_{J = \, - 1}^v} \right|,$ respectively [18]. When the ribbon width $M$ increases, the electronic gaps of both single- and bilayer ribbon structures are narrower [51], and then the distance between the two $E_{J = 1}^c$ and $E_{J = - 1}^v$ subbands reduces. This causes the shifting of peaks ${\omega _1}s$ towards a lower frequency. Furthermore, with the larger values $M,$ it is easy to notice the decrease of charge density distributed near two peaks around ${E_{\text{F}}} = 0$ of the DOS, leading to the tendency of lowering the height of the 1st peaks in the absorption spectrum. Therefore, it can affirm that the variations in the electronic properties are reflected evidently in the optical features of this group.

Besides, to seek out the regulations about the fluctuations of the strength and frequency of the 1st peaks in the optical spectra for the larger ribbon widths, we investigated the correlation between these two quantities and the ribbon width $M$ (see figure 2(d)) for both SL-AGNRs (dashed lines) and BL-AGNRs (solid lines). These outcomes reveal that the threshold frequency moves to lower energies and the height of these peaks shortens plainly when the ribbon width increases. This result is fully consistent with the consequences seen in figures 2(a) and (b). It also unveils that in the continuity of raising of $M,$ the peak position will tend to approach the frequency $\omega = 0.$ This result is in agreement with the previous observation on the increase of the metallic behaviors and eventually reaching the nature of infinite graphene sheets when enlarging the size of nanoribbons [52, 53].

Similar outcomes are obtained for the semiconducting group $3p + 1$ in both BL-AGNRs and SL-AGNRs, as shown in figure 3 for three ribbons of width $M = 4,\;7,\;10.$

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Nevertheless, by comparing the spectral structures in figures 2 and 3, we can observe a discrepancy in the spectra of two groups $3p$ and $3p + 1:$ the tendency of the moving to lower energies of the threshold frequency in group $3p + 1$ is faster than in group $3p$ (as illustrated in figures 2(b) and 3(b)). In detail, the positions of peaks ${\omega _1}s$ in BL-AGNRs for three ribbons of width $M = 4,\;7,\;10$ representing group $3p + 1$ are $0.5891{t_0},\;0.3119{t_0},\,0.2079{t_0},$ respectively (see figure 3(b)), meanwhile, the corresponding values of ribbons of width $M = 3,\;6,\;9$ within group $3p$ are $0.7624{t_0},\;0.4158{t_0},\;0.2772{t_0},$ respectively (see figure 2(b)). This discrepancy can be explained by the fact that the gap size of group $3p + 1$ is always smaller than group $3p$ for the same $p$-value [51, 54, 55]. More interestingly, for each group, the bandgap of BL-AGNRs is always smaller than that of SL-AGNRs with respect to the same dimer lines $M$ [23] resulting in the shifting of threshold frequency of BL-AGNRs to the zero-point frequency is faster than that of SL-AGNRs.

It is also worth noting that the variation of the 1st peak with the change of the ribbon width for two groups $3p$ and $3p + 1$ is similar to what was observed for the energy gap in these groups [27, 29], indicating a strong correlation between the optical and electronic properties. To reveal a correlation between the optical peaks, in particular, the first one, and the energy gap, we plot in figure 4 the relationship between the frequency of the 1st peaks and the gap size in groups $3p$ and $3p + 1$ for both SL-AGNRs and BL-AGNRs, for the same number of dimer lines $M$ computed in figures 2(d) and 3(d). Interestingly, it unveils that the position of the ${\omega _1}s$ is equivalent to the magnitude of the energy gap, ${\omega _1} = {E_{{\text{gap}}}}$ for both groups. Thus, the formation and the characteristics of the first peak depend strongly on the gap size of the structure, revealing a possibility to tune the optical properties by modulating the electronic properties, in particular, the gap size of the electronic structure.

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To present a thorough understanding of the optical spectra for all three groups of armchair ribbons, we continue to investigate the ribbons of group $M = 3p + 2.$ The results of ribbons of width $M = 5,\,8$ and $11$ are displayed in figure 5.

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Figure 5(a) points out that when ascending the ribbon width, the peak pattern of ribbons belonging to this group also exhibits many features similar to the semiconducting groups considered in section 3.1.1. The prominent absorption peaks are also distributed richly in the low energy region, and the number of peaks also augments when $M$ increases. However, figure 5(b) shows that the spectra of this group present some distinct characteristics, i.e. the 1st peak positions and intensities are almost invariable according to the ribbon width $M.$ This phenomenon is understandable due to the fact that ribbon structures of group $3p + 2$ persist in the semi-metallic state and the low-energy region is almost not modulated when changing the ribbon width [51].

Notably, the optical absorption spectra of peaks ${\omega _1}s$ for BL-AGNRs and SL-AGNRs are not identical entirely, even though groups $M = 3p + 2$ of both structures are semi-metallic. In particular, in the low-frequency range $\left[ {0,\,0.4{t_0}} \right],$ there exists a sole peak in absorption spectra of BL-AGNRs that its height and frequency position remain unchanged with respect to the change of the ribbon width. Meanwhile, this peak seems to vanish in the same region for SL-AGNRs. These consequences agree well with the alternations in the band structure of SL-AGNRs and BL-AGNRs and can be interpreted as the interlayer (van der Waals) interactions in BL-AGNRs result in the creation of extra subbands [45, 50]. In other words, the number of optical peaks in the spectral function of BL-AGNRs is more remarkable than those of SL-AGNRs.

Further analyses, we can see from the energy dispersion in figure 5(c) that the formation of the 1st peaks in group $3p + 2$ for BL-AGNRs obeys the additional selection rule that is different from the cases of the two groups $3p$ and $3p + 1.$ And this phenomenon derives from the insistence in the semi-metallic behavior of group $3p + 2$ when increasing the ribbon width [51]. This leads to the fact that subbands ${J^{\,v}} = - 1$ and ${J^c} = 1$ are almost in contact at Fermi energy for all cases of width $M.$ Consequently, the electronic excitations between parabolic bands ${J^{\,v}} = - 1$ (lying at the Fermi level) and ${J^c} = 2$ induce the presence of the optical peak in the low-frequency range. And at the same time, although the ribbon width $M$ varies, the charge distribution at these levels remains constant as reflected in the DOS panel on the right side of the band structure, and this explains the constant amplitude of peaks ${\omega _1}s$ for different $M.$

Interestingly, our further examination of the variations in the frequency and the height of the 1st peaks with the change of the ribbon widths shows that these quantities are almost unchanged with the increase of $M,$ as can be seen in figure 5(d). Thus, in the absence of electric fields, both the absorption spectra and the low-energy electronic structure of ribbons within group $3p + 2$ are nearly independent of ribbon width $M,$ concurrently the electronic properties retain in the half-metallic behavior for all ribbons.

Although the AA-stacking order is not considered herein, it is still worth noting that for the ribbon of width belonging to group $3p + 2,$ the electronic structure of AA-stacking BL-AGNRs presents a linear character around the Fermi level, the same as in the single-layer ribbon. In contrast, the low-energy bands of the AB-stacking structure are parabolic [56]. As a result, the absorption spectrum of AA-stacking BL-AGNRs is likely analogous to that of single-layer ribbons and there is no existence of the 1st peaks at low-frequency $\omega \leqslant 0.4{t_0}.$

3.2.1.1. Optical absorption spectra of two groups $M = 3p$ and $M = 3p + 1$ under the impact of a perpendicular field.

First, we consider the impact of a perpendicular electric field on the optical properties of the intrinsically semiconducting groups $3p$ and $3p + 1.$ Figure 6 presents the results corresponding to BL-AGNRs of widths $M = 6$ and $M = 7,$ representing these two groups respectively. Two cases of the applied electric potentials ${V_t} = 0.4\,{\text{V}}$ and ${V_t} = 0.8\,{\text{V}}$ were considered.

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In figures 6(a) and (b), we can observe a considerable modulation of absorption spectra when varying the field strength (via ${V_t}$). Interestingly, under the impact of the electric field, the peaks are shifted visibly, in which the subpeaks on the left-hand side of the main peak tend to move toward lower frequencies, and vice versa, those on the right-hand one shift toward higher frequencies. The insets in figures 6(a) and (b) clearly show the trend of shifting of the absorption peaks in the low-frequency region when increasing the applied voltages. Additionally, the electric field impacts significantly on the main peak at $2{t_0}$ in the ribbons of group $3p + 1.$ These phenomena are quite similar to those observed in figures 2 and 3 when varying the ribbon widths $M$ of these two groups and can be explained as follows: when enhancing the applied voltages, the displacement of subbands ${J^{c,v}} = \pm 1$ in the energy bands (see the insets of figures 6(a) and (b)) is the main reason for the moving of peaks ${\omega _1}s.$ When the electric potentials increase, these subbands tend to get closer to each other, and then the gap size is narrower as can be seen from the electronic structures in figures 6(c) and (d). Besides, we also found that the intensity of peaks ${\omega _1}s$ declined greatly at the stronger electrostatic potentials.

Therefore, the outcomes unveil that the perpendicular electric field has strong effects on modulating the frequency and the height of the 1st peaks in the spectral function for both groups $3p$ and $3p + 1.$ It is also worth noting that in the comparison of the displacement of the 1st peaks among these two groups, the results show that for the same applied voltage, the peaks ${\omega _1}s$ in group $3p + 1$ exhibit a tendency of the move to the low-frequency range faster than in group $3p.$ In detail, when applying different potentials ${V_t} = 0\,{\text{V}},\,0.4\,{\text{V}},\,0.8\,{\text{V}}$ in the ribbon of width $M = 7,$ the frequencies of the 1st peaks (R1), (Y1), (G1) are $0.3119{t_0},0.2599{t_0},0.1559{t_0},$ respectively. For the ribbon of width $M = 6,$ the values correspond to $0.4158{t_0},\,0.3465{t_0},\,0.2426{t_0},$ respectively. Such discrepancy is because the magnitude of the energy gap in group $3p + 1$ reduces formidably compared to that in group $3p$ at the same applied potential ${V_t}.$

Further analysis, in the higher-frequency region, i.e. $0 < \omega < 0.6{t_0},$ as shown in figures 6(c) and (d), we can observe three new peaks marked as (Y1ʹ), (Y2ʹ), and (G1ʹ) in both ribbon structures. These peaks are determined at the frequencies of $0.5198{t_0},\;0.5545{t_0},\;0.5545{t_0}$ (in group $3p$), and $0.4678{t_0},\,\;0.5198{t_0},\;0.4851{t_0}$ (in group $3p + 1$) when ${V_t} = 0\,{\text{V}},\,0.4\,{\text{V}}$ and $0.8\;{\text{V,}}$ respectively. These peaks do not obey the selection rule as for the 1st peaks but are satisfied with the extra selection rules [19]. In particular, the possible transitions happening between ${J^{\,v}} = - 1$ ($\pi$ bands) and ${J^c} = 2$ (${\pi ^*}$ bands) or from ${J^{\,v}} = - 2$ ($\pi$ bands) to ${J^c} = 1$ (${\pi ^*}$ bands) are illustrated by the dashed green and yellow arrows in the electronic bands in figures 6(c) and (d). This indicates an increment of transition channels when enhancing the electric field strength as the excitations not only occur from the top valence band $\left( {{J^{\,v}} = - 1} \right)$ to the bottom conduction one $\left( {{J^c} = 1} \right)$ but also with other different energy levels in the subband structure. Thereby, the reduction of the amplitude of the 1st peaks and the enlargement of the height of neighboring new peaks are seen at higher voltages. Moreover, these changes are reflected by strong alternations in the number and the position of peaks in the DOS panels, as shown in figures 6(c) and (d).

Furthermore, to provide an overall picture of the variation in the intensity as well as the position of the peaks ${\omega _1}s$ with the applied potential, we considered these quantities when varying ${V_t}$ in the range from −1 V to 1 V as shown in figures 6(e) and (f). The violet and pink lines (relative to the peak height and position) point out that these quantities reduce with the increase of ${V_t},$ in particular, a rapid reduction is observed in the range of ${V_t} = \left[ { - 0.5\,{\text{V}},\,0.5\,{\text{V}}} \right].$ The phenomenon of moving towards the lower frequencies of the peaks ${\omega _1}s$ can be elucidated by the narrowing of the gap size under the effect of electric fields [27, 29]. When the excitation energy ${\omega ^{cv}}$ reduces, electrons are easily transmitted between the valence and conduction bands, leading to the increment of the optical conductivity.

Thus, the perpendicular field impacts significantly on the position, the number, and the height of the optical peaks in semiconducting groups of BL-AGNRs.

3.2.1.2. Optical absorption spectra of group $M = 3p + 2$ under the impact of a perpendicular field.

To understand completely the optical properties of all three groups of ribbon structures under the effect of a perpendicular electric field, we continue to analyze the optical spectra of group $3p + 2,$ which is represented via the results obtained in figure 7 for the ribbon of width $M = 8.$

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Figure 7(a) shows that the perpendicular electric field gives rise to new peaks contiguous to the threshold absorption peaks as well as adjusts the shape of the main peak near $2{t_0}.$ Specifically, observing the energy bands in figure 7(b), it can be seen that the energy gap exhibits a trend of, first, opening, and then enlarging in the presence of the electric field, i.e. ${E_{{\text{gap}}}} = 0.1044{t_0},\,\,0.128{t_0}$ when ${V_t} = 0.4\,{\text{V,}}\,\,0.8\,{\text{V,}}$ respectively. It is worth mentioning that the two new peaks (Y1ʹ) and (G1ʹ) obey the selection rule $\Delta J = 0.$ Here, once again we notice a correlation between the excitation energies in the electronic band and the peak positions ${\omega _1}s$ in the spectral function. For example, for ${V_t} = 0.4\,{\text{V,}}$ the peak frequency (Y1ʹ) is positioned at ${\omega _1} \approx 0.1044{t_0}$ equal to the energy difference between the energy levels ${E^{\,v}} = - 0.0484{t_0}$ (corresponding to ${J^{\,v}} = - 1$) and ${E^c} = 0.056{t_0}$ (corresponding to ${J^c} = 1$) in the energy bands. Furthermore, when varying ${V_t},$ the two new peaks (Y1ʹ) and (G1ʹ) tend to move to higher frequencies. The above outcomes are distinct from the behaviors of groups $3p$ and $3p + 1.$

Comparing the amplitude of these two new peaks, (G1ʹ) is more outstanding than (Y1ʹ), thus demonstrating that the increase of the potential strength leads to the augmentation of the peak height. Such a change in the optical spectrum is strongly correlated with the change in the electronic DOS (on the right-hand side of the energy dispersion). We can observe the formation of the first DOS peak around $E = 0$ when applying an electric field, and with a stronger applied voltage, these two new peaks shift far away from the zero energy point. For example, these DOS peaks are determined at the energy levels ${E^c} = 0.056{t_0}$ and ${E^c} = 0.069{t_0}$ when ${V_t} = 0.4\,{\text{V,}}$ and $0.8\,{\text{V,}}$ respectively. In addition, we also realize that the splitting effect occurs in the vicinity of the degenerate levels in the DOS, implying that the perpendicular electric field has a vigorous impact on the modification of the sub-energy bands of BL-AGNRs.

We also examined clearly a variation of the position and the height of peaks (R1), (Y1), and (G1), as shown in figure 7(c) for the range of frequency $0.14{t_0} \leqslant \omega \leqslant 0.4{t_0}.$ In which, the dashed arrow in each inset panel of the energy structure illustrates the optical transition of these peaks corresponding to the applied potentials ${V_t} = 0\,{\text{V}},\,\,0.4\,{\text{V}},\,0.8\,{\text{V}}{\text{.}}$ It is worth mentioning that these excitation channels have the same regulation, arising from the energy level ${J^{\,v}} = - 1$ to the one ${J^c} = 2$ in the electronic structure. For instance, the transition from ${E^{\,v}} = - 0.077{t_0}$ to ${E^c} = 0.1599{t_0}$ causes the formation of the peak (Y1) at $\omega \approx 0.2369{t_0}.$ Notably, several Mexican-hat shape dispersions could appear when the applied voltage is large enough as seen in the band structures in figures 7(b) and (c). This phenomenon is explained by 'the band inversion', as presented in [57, 58].

To understand the variations in the amplitude and the position of the first peak in the optical spectra, we plotted these quantities as a function of the applied potential in the range $\left[ { - 1\;{\text{V}},\;1\;{\text{V}}} \right],$ as shown in figure 7(d). The results show that these two quantities seem to augment significantly in the range $\left[ { - 0.5\;{\text{V}},\;0.5\;{\text{V}}} \right],$ then beyond this region, the rate of increase tends to be slower. In other words, when the gap size is extended vividly in the small values of the applied voltages, the position of the peaks ${\omega _1}s$ reaches a higher frequency. This result is consistent with the previous studies [27, 29], and the above discussions.

The above analyses indicate vigorous effects of a perpendicular electric field on the absorption spectra. This could lead to the question of how physics changes under the impact of a transverse electric field. To answer this questionnaire, we keep investigating the variation in the spectral function of BL-AGNRs as well as SL-AGNRs under the presence of a transverse electric field. The results are shown in figure 8.

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Observing the low-energy absorption spectra in figures 8(a)–(c) for various ribbons of width $M = 6,\,7,$ and $8,$ we notice that the features of the optical structure in BL-AGNRs and SL-AGNRs do not change much compared to the case without the external fields. Specifically, the outcomes show that there is no emergence of any striking peaks in the absorption spectra when varying the field strengths, meaning that the transverse electric fields do not enrich the optical properties as in the case of the perpendicular ones. The only tunability is the reduction in the height of the main peak at $\sim 2{t_0}$ in the rising of the applied voltage. As analyzed in figure 2(a), to form this peak, electrons are excited among the parabolic bands ${E^{c,v}}\sim \pm {t_0}$ (see figures 8(a) and (c)), or the complete flat bands ${E^{c,v}} = \pm {t_0}$ (see figure 8(b)). The electrostatic potentials lead to a redistribution of electrons in these subbands, causing variations in the height of the main peak as well as the small perturbations of subpeaks nearby.

Importantly, the position of the peaks ${\omega _1}s$ seems to be invariant in raising the applied potential ${V_s}$ for all the considered ribbons. Such a result can be interpreted as the bandgap is not opened or the excitation energy among the subbands adjoining the Fermi level to form these peaks is almost constant (indicated in the inset, on the left-hand side of each absorption spectrum). Moreover, when examining the low-energy absorption spectra of SL-AGNRs, as indicated in the inset in figure 8(c), we recognize that there does not exist any 1st peak in the peak structure of the metallic group $\left( {3p + 2} \right)$ in the low-frequency region.

In figures 8(d) and (e), we examine the dependence of position and amplitude of the peaks ${\omega _1}s$ on the applied voltages ${V_s}$ in the range $\left[ { - 1\,{\text{V}},\,1\,{\text{V}}} \right]$ for SL-AGNRs (groups $3p$ and $3p + 1$) and BL-AGNRs (groups $3p,\;3p + 1$ and $3p + 2$). The results disclose that the frequency position of the 1st peaks in each group is nearly constant in the considered range of the applied potential ${V_s}.$ Nevertheless, the position of these peaks in different groups and distinct structures is not the same. Besides, we also denote that the height of these peaks is more sensitive under the impact of transverse fields (see figure 8(e)), in which, group $3p$ of BL-AGNRs has the most considerable changes among the considered structures.

In the analyses for the impacts of the transverse and perpendicular electric fields, we have thus seen the dissimilarities in modulating the optical and electronic properties of each field. The perpendicular field exhibited a strong impact on the gap size and electron distribution, resulting in significant changes in the increment of the number, the alteration of the height, and the position of the optical peaks. On the contrary, the transverse field presented weak impacts on these quantities. Such a discrepancy between the effects of two electric fields on the optical properties is similar to what is observed in the electronic properties [27, 29]. And the distinguished influences can be explained as follows: in BL-AGNRs, the perpendicular field generates the same potential at the atomic sites in the same plane, and therefore its impact does not rely on the ribbon width $M.$ Hence, the influence of this type of electric field is characterized mainly by the applied potential ${V_t}.$ In contrast, with the model of applying the transverse field, the atoms are lying away from the electric gating $- {V_s}/2$ (chosen as the origin) with various distances, meaning that the effect of this field is dependent directly on the ribbon widths. Interestingly, the perpendicular field exhibits a more strong influence than the transverse one in the alternation of the peak structure for optical spectra of the AB-stacking sequence.