Role of Size-Dependent Damping Due to Electron–Surface Scattering on the Al Nanoparticle-Based Deep Ultraviolet Surface-Enhanced Fluorescence

Herein, we report the theoretical investigation to understand the role of size-dependent damping (SDD) due to electron–surface scattering on the Al nanoparticle-based deep ultraviolet surface-enhanced fluorescence. First, the absorption spectra and electric field enhancement (EFE) inside and outside Al nanoparticles of different sizes are plotted without and with considering SDD. Later, the role of SDD on the near- and far-field plasmonic properties of Au and Ag nanoparticles of different sizes is investigated for comparison. Finally, Al nanoparticle-based SEF enhancement is estimated for different nanoparticle sizes, emission wavelengths, and separations between nanoparticle and fluorophore without and with considering the SDD.


Introduction
Surface-enhanced fluorescence (SEF) has recently become a valuable tool for several biological applications. Conventionally, the SEF technique relays on the functionalization of the bioanalytes, such as viruses/protein molecules with fluorescence tags or labels [1][2][3][4][5]. However, functionalization is a tedious task and needs a few additional chemicals. Importantly, the label-based SEF technique belongs to one of the destructive techniques. In contrast, the deep ultraviolet (UV) SEF does not depend on the functionalization of bioanalytes because it involves the enhancement of intrinsic fluorescence from bioanalytes that falls in the deep UV region (where radiation with a wavelength shorter than 350 nm) [6]. For deep UV-SEF, there is a need for metal or plasmonic nanoparticles that have localized surface plasmon resonance (LSPR) in the deep UV region. Even though the Au and Ag nanoparticles are quite popular in the field of nanoplasmonics, these cannot be used in the deep UV region due to the predominant interband transitions [7][8][9][10][11][12]. Fortunately, the LSPR of Al nanoparticles exists in the deep UV region; therefore, these nanoparticles are helpful for the deep UV-It is well-known that the Al exhibits metallic character in the broader wavelength range relative to the Au and Ag, and it has a high density of free electrons and a conduction band with 3 electrons per atom (as opposed to 1 electron per atom for Au and Ag) [13]. Al is less expensive and more readily available than noble metals because it is a relatively common substance on earth. This is a fantastic advantage for several applications relating to the industry, which might have a significant effect in terms of economic gains [14]. The Al nanoparticles are also found useful for several other applications such as UV-SEF [15], label-free biosensing [16], non-linear plasmonics [17,18], light-harvesting devices [19], photodetection [20], photocatalysis [21,22], high data density storage [23], optoelectronics [13], selective photodetoxification [24], and making efficient photoelectrodes [25]. The Al nanoparticles can be prepared using different methods such as lithography technique [26], electrochemical method [27], laser ablation technique [28], and wet-chemistry method [29,30].
Researchers have reported deep UV-SEF enhancement using isolated [16] and coupled [16,31] Al nanoparticles of different sizes. The Al-Ag bimetallic nanoparticles are also found useful for enhancing UV fluorescence [32]. In general, the SEF enhancement ( SEF ) strongly depends upon the excitation wavelength ex , local electric field enhancement (LEFE), and fluorescence or emission wavelength em . Recently, the theoretical values of SEF and LEFE are observed to be maximum for small nanoparticles that support only the dipole mode [16]. The same has been proved even in the case of Al nanoparticles in the present study. It is important to note that the SDD due to the electron-surface scattering [11] becomes predominant in the small Al nanoparticles. In order to have a proper comparison between experimental and theoretical values, the size-dependent dielectric function [33] of Al must be taken into account while plotting the LSPR spectra and determining the values of LEFE and SEF . However, to the best of our knowledge, the effect of SDD in Al nanoparticles on the LSPR spectra, LEFE, and UV-SEF enhancement is not reported in the literature. This inspired us to carry out the current theoretical investigation.
Herein, we report (i) the role of SDD on the absorption spectra and LEFE of spherical-shaped Al nanoparticles of different sizes, (ii) the role of SDD on the absorption spectra and LEFE of Au and Ag nanoparticles of different sizes for comparison, and (iii) the dependence of SEF on the separation between the fluorophore and nanoparticle (d), em , and ex in the absence and presence of the SDD in Al nanoparticles.

Theoretical Aspects
According to the electromagnetic theory, the SEF can be estimated using the following expression [34,35]: where the first term in the middle expression represents the excitation rate enhancement and the second term represents the quantum yield enhancement. The rightmost expression represents the local electric field intensity enhancement. All these terms can be estimated using the analytical expressions given in Refs. [36][37][38]. It is important to note that for estimating the LEFE and SEF of the nanoparticles, one should know the wavelengths of the plasmon modes. One can easily obtain these wavelengths either from the absorption or scattering spectra of single nanoparticles [39][40][41]. Since the lower-and higher-order modes are relatively prominent in absorption spectra rather than the scattering spectra [6], in the present study, we have plotted the absorption spectra (absorption efficiency ( Q abs ) versus wavelength plots) of single Al nanoparticles of different sizes using the analytical expression given in Ref. [38]. After obtaining the wavelengths of the plasmon modes, the electric field enhancement (EFE) distribution inside and outside the Al nanoparticles is plotted to estimate np ex ∕ 0 ex . Finally, the SEF is estimated by varying the ex , em , and d.
The absorption spectra, LEFE, and SEF are strongly dependent upon the dielectric function of the Al nanoparticles. The analytical expression for the complex dielectric function of the Al is given as follows [33]: where f ( ) is the intra-band part of ( ) described by the well-known free electron or Lorentz-Drude model. Similarly, b ( ) is the interband part of ( ) described by the simple semi-quantum model resembling the Lorentz result for insulators and k is the number of oscillators with frequency j , strength f j , and lifetime 1∕Γ j , while Ω p = √ f 0 p is the plasma frequency associated with intra-band transitions with oscillator strength f 0 , and p is the plasma frequency and damping constant Γ 0 . The values of f j , j , and Γ j are taken from Ref. [33].
It is crucial to note that in small nanoparticles, the plasmon damping due to the electron-surface scattering is predominant [42][43][44]. Therefore, the ( ) becomes size-dependent. This means, in Eq. (2), the value of Γ 0 needs to be replaced by Γ d (R) , and its expression is given as follows [45]: Here, F is the Fermi speed of free electrons (= 2.0 × 10 6 m/s) in Al. R eff is the effective mean free path of the electrons, and it is equal to the radius of the nanosphere (R) [45]. The typical value of α is 1 for isotropic scattering [46]. h is the Planck's constant. From Eqs. 2 and (3), it is clear that the electric permittivity gets modified significantly in small nanoparticles in the presence of SDD. Due to this, the absorption spectra, LEFE, and SEF are expected to be changed. Therefore, in the present work, we have also estimated the values of , , and which are defined as follows: Maximum absorption ef f iciency in the absence of SDD Maximum absorption ef f iciency in the presence of SDD Here, FWHM and max represent the full-width at halfmaximum and maximum LEFE, respectively. Figure 2a shows the absorption spectra of single Al nanoparticles of R varying from 5 to 20 nm. The peaks in these spectra represent the lower-and higher-order plasmon modes. Only the dipole mode is present in the case of the Al nanoparticles with R = 5 nm. One weak quadrupole mode is observed when the R value is increased from 5 to 10 nm, and it is prominent when the R value is increased further. It is important to note that the higher-order modes are excited even in small Al nanoparticles as compared with Au, Ag, and, indium nanoparticles [6,47,48]. This can be understood as follows. The plasmon modes of gold and silver nanoparticles exist in the visible region. However, the plasmon modes of small Al nanoparticles exist in the deep UV region (i.e., shorter wavelength region). It is well-known that the phase change of the light across the nanoparticles is very much dependent upon the wavelength. A significant phase change across small Al nanoparticles is expected in the deep UV region. The phase change is the origin of the higher-order plasmon modes. In Fig. 2a, the plasmon modes are getting red-shifted upon increasing the R value, and especially, the lower-order modes are expected to be present in the visible region for a large R value. In the case of small nanoparticles, all the observed modes are present in the UV region. Therefore, the small nanoparticles are useful for enhancing the UV-SEF. However, in the case of small nanoparticles, the plasmon damping due to electron-surface scattering is predominant as mentioned above. Therefore,  one has to consider the size-dependent dielectric function for plotting the spectra of single nanoparticles Fig. 1. Figure 2b represents the modification of the absorption spectra of single Al nanoparticles after considering the SDD. The modes are broadened after considering SDD. As expected, the SDD does not have a significant role in the wavelengths of the plasmon modes. The maximum Q abs and FWHM of the modes in the absence and presence of the SDD are significantly different when the R is lower than 10 nm. However, the difference between these values is found to decrease upon increasing the R value ( Fig. 2c  and d). For comparison, the absorption spectra of single Au and Ag nanoparticles are also plotted without and with considering the SDD (Fig. S1 in the supplementary information). From all the spectra, the FWHM values of the plasmon modes are estimated and shown in Fig. S2a in the supplementary information, and from this panel, the vs. R curves are plotted for Au and Ag and shown in Fig. 2e. From  Fig. 2e, It is clear that the SDD is playing more role in Ag nanoparticles and less role in Al nanoparticles. From Fig. 2f, we can conclude that the effect of SDD on the FWHM of the plasmon modes is significant at lower R values. Figure 3a shows the absorption spectra of large Al nanoparticles without SDD. As expected, the number of higherorder plasmon modes in Fig. 3a is more relative to those observed in Fig. 2a. Also, the number of higher-order modes is increased with the R value. However, the amplitude of the modes is found to decrease upon increasing R value. Upon increasing the R value, the lower-order mode is red-shifted, and the higher-order modes exist in the lower wavelength region as in the case of Au and Ag nanoparticles. In the large Al nanoparticles, the prominent higher-order modes exist in the deep UV region, and these modes play a major role in the large Al nanoparticle-based deep UV-SEF. Figure 3b shows the absorption spectra obtained after considering the SDD. After considering the SDD, there is no significant change in the number, wavelength locations, and FWHM of the plasmon modes. This means the SDD can be ignored in the large Al nanoparticles.

Effect of SDD on the LEFE Distribution of Single Al Nanoparticles
As mentioned in the theoretical section, the excitation rate enhancement of the fluorophore kept in the vicinity of a metal nanoparticle is directly related to the LEFE. Therefore, the electric field enhancement (EFE) distribution inside and outside of single Al nanoparticles of different R values is plotted without and with considering the SDD. Figure 4 shows the EFE distribution estimated at the wavelengths of plasmon modes of a few small Al nanoparticles. Panels a, c, and e represent the dipole modes in the case of Al nanoparticles with R = 5, 10, and 13 nm, respectively. The EFE is found to change significantly in these nanoparticles in the presence of the SDD (panels b, d, and f). However, there is no change in the mode pattern after considering the SDD. The higher-order modes start observing from R = 13 nm. However, the EFE observed at the wavelength of the higherorder plasmon mode in the case of R = 13 nm is significantly lower than that observed at the wavelength of the dipole mode. The EFE values observed at the higher modes are larger when the R value is more than 13 nm. From all panels of Fig. 4, the LEFE is maximum at the wavelengths of dipole modes of the smaller nanoparticles, and the incident light's phase is essentially constant across these nanoparticles, forcing every free electron to oscillate coherently. Fig. 3 a and b show the absorption spectra of large Al nanoparticles (of different R values) obtained without and with considering the SDD, respectively. Small portions of the spectra are zoomed in and shown as insets for better clarity. For all cases, the value of n med is fixed at 1.33

3
To obtain the optimum LEFE, the graphs between the max obtained for the wavelengths of the dipole and quadrupole modes versus R graphs are plotted and shown in Fig. 5a, b, respectively. From panel a, it is apparent that the max is optimum when R = 10 nm in the absence of the SDD. However, after considering the SDD, the optimum max is observed for R = 13 nm. This deviation could be due to the modification of the dielectric function in the presence of the SDD. The optimum max in the presence of SDD is reduced by a factor of 5 possibly due to an increase in the imaginary part of the dielectric function. From panel b, it is apparent that the max values are significantly lower as compared to those mentioned in panel a. Here also, the obtained values after considering the SDD are significantly lower Fig. 4 The EFE distribution of single small Al nanoparticles excited at the wavelengths of lower-and higher-order plasmon modes. a, c, e,  g, i, k, m, o, q, and s are obtained without considering the SDD and   b, d, f, h, j, l, n, p, r, and t are obtained with considering the SDD. For all cases, the value of n med is fixed at 1.33 in the presence of the SDD. In panel a, the max is found to decrease upon increasing the R value beyond 13 nm due to the suppression of the dipole mode, but it is the opposite in panel b, and this indicates the progression of the quadrupole mode upon increasing the R value beyond the 13 nm.
For comparison, the EFE distribution inside and outside of single Au and Ag nanoparticles is also plotted without and with considering the SDD, and the max versus R curves are shown in Fig. S2b in the supplementary information. From this panel, the vs. R curves are obtained for Au and Ag and shown in Fig. 5c. The shape of the vs. R curve obtained for Al is significantly different compared to that obtained for Au and Ag nanoparticles, even though the shape of the versus R curves is the same for all these nanoparticles. Here also, the SDD is playing more role in Ag nanoparticles relative to the Au nanoparticles. From Fig. 5d, it is clear that the values of obtained for the excitation at the wavelengths of the quadrupole modes of Al nanoparticles are found to be close to 1. This means the SDD does not influence the higher-order plasmons modes significantly.
The EFE distribution is also plotted at the wavelengths of all plasmon modes of large nanoparticles without and with considering the SDD. For these calculations, we have considered the modes that exist only in the deep UV region. The EFE distribution at the wavelengths of a few plasmon modes observed in the large nanoparticles is shown in Fig. 6. The max values observed at the wavelengths of all higher-order modes exist in the deep UV region are shown separately in Table 1. From Fig. 6, it is clear that the role of SDD on the EFE and mode pattern is insignificant.

Role of SDD on SEF
From Eq. (1), it is clear that the SEF is also directly proportional to the quantum yield enhancement which is strongly dependent on the radiative rate enhancement ( r ∕ 0 r ) and nonradiative rate enhancement ( nr ∕ 0 nr ). The values of r ∕ 0 r and nr ∕ 0 nr are the functions of the dielectric function of the Al nanoparticles. Therefore, the quantum yield enhancement is also expected to be modified in the presence of SDD as the LEFE or excitation rate enhancement. Figure 7 shows the direct comparison between the SEF in the absence and presence of the SDD. From all panels, we can conclude the following points. The small nanoparticles are useful only for UV-SEF because the value of SEF is very low when the em falls in the visible region. The SEF values are found to decrease significantly after considering the SDD except for R = 13 nm. The maximum value of SEF ( max SEF ) is observed when there is a little separation between fluorophore and nanoparticle as in the case of Al, Au, and indium nanoparticle-based SEF.
To find out the optimum enhancement, we have estimated the max SEF for different Al nanoparticles excited at the wavelengths of the dipole and quadrupole modes, and the estimated values are shown in Fig. 8. The maximum values of SEF are observed for nanoparticles with R = 10 nm and 13 nm in the  Table 1 Effect of SDD on the higher-order plasmon mode wavelengths, max and max SEF values in the case of large Al nanoparticles. Here, the parameters with the subscripts "bulk" and "SDD" represent the values obtained without and with SDD in the large Al nanoparticles, respectively absence and presence of SDD, respectively. As mentioned in the above section, the max or maximum excitation rate enhancement is observed for the nanoparticles with R = 10 nm and 13 nm in the absence and presence of SDD, respectively. This clearly indicates that the excitation rate enhancement is contributing more to the max SEF as compared with the quantum yield enhancement. The max SEF is in the order of 10 3 in the absence and presence of the SDD in the nanoparticles excited at the wavelengths of the dipole modes. However, the max SEF values are found to be reduced by 1-2 orders of magnitude when the Al nanoparticles are excited at the wavelengths of the quadrupole modes. Finally, one can conclude that the reduction in the max SEF is observed in the presence of the SDD in both panels. Table 1 shows the wavelength locations of the higherorder modes in the deep UV region and the corresponding max and max SEF obtained without and with considering the SDD. From this table, we can conclude that the role of SDD on the plasmon mode wavelengths and max values is insignificant. However, there is a significant variation in the max SEF values after considering the SDD in the large Al nanoparticles mainly due to the modification of quantum yield enhancement as per Eq. (1). It is to be noted that the variation in max SEF values is not systematic. Because the max SEF values obtained for a few plasmon mode wavelengths are larger in the absence of the SDD, for few plasmon mode wavelengths, the opposite trend is observed.

Conclusions
We have successfully studied the role of SDD due to the electron-surface scattering on the (i) absorption spectra and LEFE of single Al nanoparticles of different R values and (ii) Al nanoparticle-based deep ultraviolet surface-enhanced fluorescence. The significant higher-order plasmon modes are observed even in small Al nanoparticles. The effect of SDD on the FWHM and intensity of lower-order mode (dipole mode) are prominent relative to the higher-order mode. Among Au, Ag, and Al nanoparticles, the variation in the FWHM and intensity of the plasmon modes are more in the Ag nanoparticles and less in Al nanoparticles. The max SEF is observed in the order of 10 3 in the absence and presence of SDD in the Al nanoparticles. This means that by using Al nanoparticles, one can enhance the intrinsic deep UV-SEF of single-protein molecules up to 3 orders of magnitude. Therefore, the small Al nanoparticles are efficient for UV-SEF applications.

Availability of Data and Materials
The data that support the findings of this study are available from the corresponding author upon reasonable request.

Declarations
Competing Interest The authors declare no competing interests. Fig. 8 a and b show the role of R and SDD on the max SEF obtained for Al nanoparticles excited at the wavelengths of dipole and quadrupole, respectively