Average enhancement factor (AEF) of a coreshell (Ag@SiO2) on the fluorescence of molecules doped within the silica shell is proposed and studied to estimate the overall performance of a large number of coreshells. Using Mie theory and dyadic Green’s functions, the enhancement factor (EF) of a coreshell is first calculated for any arbitrarily oriented and located electric dipole embedded in the shell. AEF is then obtained by averaging the individual EF over all possible orientations and positions of the electric dipoles. AEF of a FITC-doped coreshell (radius of Ag core: 25 nm, thickness of shell: 15 nm) irradiated by a laser of 488 nm for FITC’s emission at 518 nm is 2.406. It is much smaller than the maximum EF (30.114) of a coreshell containing a single molecule with a radial orientation at its optimal position. For Alexa 430-doped coreshell excited at 428 nm, AEF is 12.34 at the emission of 538 nm.
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Recently, numerous metallic nanostructures have been developed for the applications of surface-enhanced fluorescence (SEF) [1–8] (or called metal-enhanced fluorescence ) and surface-enhanced Raman spectroscopy (SERS) [10,11] by utilizing the surface plasmon resonance (SPR) effect of metals. Several studies showed that the enhancement factor (EF) of a metallic nanoparticle (MNP) on the fluorescence of a single molecule depends on the orientation and the location of the molecule relative to the MNP, the polarization of the incident light , and the distance from the molecule to the MNP [13,14]. Moreover, there are several other factors affecting EF, e.g. the size of the MNP  and the overlap of the absorption and the emission spectra of the molecule with the SPR band of the metal  etc. On the other hand, there are a few experimental researches focused on the overall performance (enhancement or quenching) of a large number of MNPs on a large amount of molecules [3,6], rather than a single MNP on a single molecule. In order to confine the molecules in the proximity of a MNP with a preciously controlled location, coreshell (Au@SiO2 and Ag@SiO2) structures [3–6] were then developed, where metal cores were coated by molecules-doped silica shells. Sometimes, a silica spacer, in between the molecules-doped shell and the metal core, is used to separate the molecules and the metal in order to avoid the quenching effect, which is due to a too short distance resulting in a strong energy transfer.
There seems to be a large discrepancy in the enhancement factors (EFs) of previous experimental [3–6] and numerical [12–14] results. The results of numerical simulations usually present much larger EFs than their experimental counterparts. The discrepancy is probably due to the fact that numerical simulations usually consider only some special configurations of the MNP and the molecule, while overlook the random orientations and locations of molecules with respect to the MNP in the experimental setup. Therefore, a reasonable estimation for the overall performance of a larger number of molecules-doped coreshell on fluorescence by numerical method is extremely needed for an optimal design and further applications.
In order to identify the EF of a large number of molecules-doped coreshells, we propose an idea of average enhancement factor (AEF) by theoretically considering the effects of all possible orientations and locations of these molecules in the shell. The theoretical work is based on the analytical solutions of Mie theory of a plane wave  interacting with a spherical coreshell, as well as dyadic Green’s functions [16,17] of an electric dipole (embedded in the shell) interacting with the coreshell. Using these solutions, the individual EF of a coreshell with any arbitrarily oriented and located molecule is calculated. The AEF is then obtained by averaging the individual EF over all possible orientations and positions of the molecules.
Figure 1 shows the configuration of a molecule (an electric dipole) embedded in the silica shell of a coreshell irradiated by an incident polarized EM wave. The radii of the Ag core and the coreshell are denoted byand, respectively, and the thickness of the silica shell is . Without loss of generality, the incident wave is assumed to be x-polarized and propagate in the z-direction. The origin of the coordinate system locates at the center of the coreshell. The multi-scatterings of light among multi-coreshells are neglected for a dilute colloid. In addition, the interaction of multi-molecular fluorescence is also neglected.
2.1 Excitation rate
When an incident plane wave illuminates a spherical coreshell, a strong local electric field is induced in the proximity. Since the molecules are doped within the silica shell, we use the Mie theory [10,17] of a multi-layered sphere to calculate the electric field in the shell. The intensified field can raise the probability for exciting molecules in the near field. The excitation rate for a molecule at a specific excitation wavelengthis defined as, where xd and ep are the position vector of the molecule and the unit vector of the molecular dipole moment, respectively, and the denominator is the intensity of the incident wave.
2.2 Apparent quantum yield
Once the molecule is excited, it behaves as an oscillating electric dipole for the following emissions. Since the dipole is in the proximity of a MNP, its radiative and nonradiative decay rates will be affected dramatically by the SPR of the MNP. The dipole moment vector ep can always be decomposed into its radial (normal) and tangential components. In order to systematically analyze the field induced by an arbitrarily oriented dipole interacting with a coreshell, two sets of dyadic Green’s functions are analytically derived by using the method of Ref ; they are for a unit radial dipole and a unit tangential dipole, respectively, located within the shell. The EM fields induced by these two types of dipoles are in series forms of spherical wave functions at an emission wavelength. The total EM fields generated by an arbitrarily oriented dipole in the presence of the coreshell are then the linear combinations of the EM fields of these two sets of analytical dyadic Green’s functions. With these EM fields at hand, the radiative decay rate and the nonradiative decay rate of the dipole in the presence of the coreshell can be computed by evaluating the corresponding surface integrals of the EM field [18,19]. Furthermore, the apparent quantum yield η, which is a function of xd, ep, and the emission wavelength, is defined as .
2.3 Enhancement factorEq. (2) to estimate the MEF of the surrounding medium on the fluorescence of a single free molecule stimulated by a polarized light in the absence of the coreshell. Since the apparent quantum yield is unit one, for the case without coreshell, its MEF is then 1/3.
The effective enhancement factor (EEF), , is defined as the ratio of the MEF of a coreshell in the surrounding medium to the MEF of the medium without the coreshell, in order to take further into account the relative effect of a surrounding medium with and without a coreshell on a single molecular fluorescence. Therefore the EEF of a coreshell on the fluorescence at position xd excited by a polarized light of wavelength and emitting a fluorescence of wavelength, is .
If the molecules are uniformly distributed on the surface of a spherical layer with a distance d to the spherical core, the average enhancement factor (AEF), , of a coreshell containing an infinitely thin molecules-layer in the shell with a constant distance d away from the metal core is defined asEq. (4), the integral is integrated from the inner radius of the molecules-doped layer to the outer radius of the layer, and . Here is the thickness of the inner spacer to keep the molecules away from the metal core. If there is no spacer, then, and . It is worthwhile to mention that, for certain isotropic emitters (e.g. quantum dots ; QDs), there is no orientation dependence for the excitation rate. Hence, Eqs. (1) and (2) need to be modified for this type of isotropic emitters, and the AEF can still be applied to evaluate the overall performance of QDs-doped coreshell structure.
3. Numerical results and discussion
Several studies have shown that the EF of Ag@SiO2 on a single molecule depends on several conditions . On one hand, it is sensitive not only to the molecular location with respect to the coreshell and the incident direction of the illuminating light, but also to the dipole’s orientation relative to the polarization of the illuminating light. On the other hand, the EF of Ag@SiO2 is also dependent on the excitation and the emission spectra of the fluorescent molecule with respect to the SPR spectrum of Ag NP. In order to illustrate the frequency-dependence of AEF of Ag@SiO2 on the molecular fluorescence, we will discuss cases with and without Stokes shift, separately. In the following calculation, the coreshell is assumed to be submerged in a surrounding medium (water), and the frequency-dependent permittivity of silver of Ref . is adopted.
3.1 Non-Stokes shift
In this section, we assume there is no Stokes shift between the excitation and the emission spectra of molecules; i.e. . Consider a typical coreshell () containing a single molecule embedded in the middle of the silica shell. Four typical molecule positions (A, B, C and D) are considered, which locate on x, −z, y and z-axes, respectively, but have the same distance (d = 7.5 nm) to the surface of the Ag core, as shown in Fig. 1. When the coreshell is irradiated by a x-polarized light of wavelength propagating in the z-direction, the 3D profile of the EF for an electric dipole versus dipole orientations at the position A is depicted in a spherical plot as shown in Fig. 2(a) , where the EEF, , is 55. The x-z, y-z, and x-y plane cross sections of this profile are shown in Figs. 2(b), 2(c), and 2(d), respectively. In this dumbbell-shape profile, the maximum EF of the position A occurs when the dipole orients itself along the x-axis, parallel to the polarization of the incident light.
To illustrate that the EEF is sensitive to the relative position, let us consider molecules located at a fix distance (d = 7.5 nm) to the surface of the Ag core. The 3D profile of the EEF versus the angles of molecular positions at is depicted in a spherical plot as shown in Fig. 3(a) . The x-z, y-z, and x-y plane cross sections of this profile are shown in Figs. 3(b), 3(c), and 3(d), respectively. The x-y plane cross section is the distribution of the EEF along the equator of the coreshell. Both positions A and C are then on the equator. As shown in Fig. 3(d), however, the EEF (7.512) of the position C is much less than EEF (55.052) of the position A. This is due to the fact that the incident wave is x-polarized, so the SPR oscillation makes the electric field at the position A much stronger than that at the position C. Figures 3(a) and 3(c) indicate also that the minimum EEF (6.281) occurs at the position D. Notice that the maximum EEF (56.55) occurs at the position A′ making an angle 10° from the position A, instead of right at the position A, as shown in Fig. 3(b). This finding is in agreement with Ref . According to Eq. (3), the AEF of Fig. 3(a) is 23.485 for d = 7.5 nm at .
Figure 4 shows the AEF, , versus d which is the distance between the molecules-doped layer and the Ag core of a coreshell () at . The AEF of the entire shell is 18.78, according to Eq. (4), for the case of. Figure 4 illustrates that the maximum AEF (34.37) occurs at d = 4 nm; while when d < 1 nm, the AEF is less than 1. It suggests that a quenching phenomenon  will be caused for those molecules in this region. It also indicates that, for a fixed layer thickness, using a proper spacer [3,6] in between the molecule-doped silica layer and the Ag core (say ) will help to raise the AEF of Ag@SiO2.
For simplicity, in the following discussion we will focus only on cases without spacer, i.e . Figure 5(a) shows the AEF, , versus wavelengths for a coreshell of with different shell thicknesses calculated by using Eq. (4). The results indicate that the optimal thickness of the silica shell is about 10 nm to have maximum AEF, which agrees with the results of Ref [4,5]. The size effect of the Ag core on the AEF versus wavelengths is shown in Fig. 5(b), for a coreshell of with different core radii. Figure 5(b) indicates the smaller the Ag core, the narrower the spectrum of AEF is. Moreover, there is an optimal, about 20 nm, for obtaining a narrowband (400-430 nm) AEF with a maximum peak at 420 nm. On the other hand, if the radius of the Ag core is larger than 40 nm, the AEF spectrum of coreshell is partitioned into two different bands: a narrow shorter-wavelength band (shorter than the SPR band of an Ag NP) and a broad longer-wavelength band. However, for a larger Ag@SiO2, the values of AEF in both bands become much smaller, as shown in Fig. 5(b). Therefore, from the aspect of AEF, the wavelength-selective property of a molecules-doped Ag@SiO2 highly depends on the size of the Ag core.
3.2 Stokes shift
Since every specific type of molecule has its own unique excitation and emission spectra, the Stokes-shift effect of these two spectra on the AEF needs to be further identified. For simplicity, again, we focus only on cases without spacer, i.e. , . Using Eq. (4), the AEF versus the emission wavelengths for a coreshell ofand irradiated by several different excitation-wavelength lasers ( = 405, 428, 458, 488, 561, and 633 nm) are shown in Fig. 6 . It is easily to see that, the AEF of is much higher than the others. This is due to the fact that = 428 nm is near the peak of SPR (420 nm) of the Ag core. Therefore, Ag@SiO2 has a high selectivity for enhancing molecular fluorescence. For example, for the case of Alexa 430 dye-doped Ag@SiO2, the AEF is 12.34 at , whereas for the case of CYe-doped one, the AEF is 5.046 at . Notice that the AEF is much lower than the maximum EF of a coreshell containing a single molecule with a radial oriented dipole moment at its optimal position; e.g. for FITC-doped one, the AEF is only 2.406 at, but the maximum EF is 30.114 at the position A′.
Moreover, the Stokes-shift effect of molecular fluorescence spectra on the AEF is pronounced only when the molecular excitation (absorption) spectrum overlaps the SPR band of the Ag NP; the variation of the AEF versus is relative large for and 405 nm, compared to the other excitation wavelengths. If is away from the peak of SPR of the Ag NP, not only the value of the AEF is decreased, but also the dependence of the AEF on becomes smaller; e.g. for, the curve of the AEF versus is almost flat, and the value is as low as 1.0. Therefore, due to the narrowband character of the AEF, a molecules-doped Ag@SiO2 is a highly molecule-selective nanostructure for fluorescence enhancement. For those molecules, whose excitation spectrum is red-shifted from the SPR band of the Ag NP, a larger Ag core is needed to provide a broadband AEF to cover their excitation spectrum for enhancing the fluorescence. Otherwise, a quenching, rather than an enhancement, could be obtained for the molecular fluorescence by using Ag@SiO2. For example, a larger Ag@SiO2 ( ) can still provide a 2.85-fold AEF for Alexa 647 dye at , which is stronger than 1.02-fold AEF of a smaller one ( ). This size-dependent effect is in accordance with the previous experimental results of those fluorophores emitting red-color or even NIR fluorescence [4,5] enhanced by using a larger Ag@SiO2.
To verify our model, the experimental data of Ag-Au@SiO2  on the fluorescence of a specific molecule, , is adopted for comparison. In Ref , Ag-Au (4:1) alloy was synthesized as the core, and then coated with a 5 nm thick of silica layer, where a monolayer of molecule was attached on the outer surface of silica. Since has a large Stokes shift with an excitation peak at 450 nm and an emission peak at 620 nm, the self-quenching between nearby molecules can be neglected. Because the frequency-dependent permittivity of the Ag-Au alloy is not known, the permittivity of Ag  is adopted here for calculation. Figure 7 shows the theoretical AEF, , of d = 5 nm by Eq. (3), compared with the experimental data for different a2. The theoretical AEF is not only in the same order of magnitude but also roughly in good accordance with the experimental data. The error bars of the experiment come from the distribution of the size of Ag core, which the discrepancy between the theoretical and experimental data may be attributed to.
For a molecule-doped coreshell (Ag@SiO2), EF on molecular fluorescence strongly depends on the molecular orientation and location. AEF has been proposed by considering all possible orientations and locations of molecules. We have demonstrated that AEF is more reasonable and useful than EF to evaluate the performance of a large number of coreshells on molecular fluorescence irradiated by a polarized or unpolarized light. In fact, AEF is much smaller than the maximum EF of a coreshell for a single molecule with a radial orientation at an optimal position. For Ag@SiO2, AEF is narrowband, if . This is to say Ag@SiO2 is a molecule-selective fluorescence enhancer; if the excitation and emission spectra of the molecule overlap the SPR band of the Ag core (≈420 nm), the maximum AEF is obtained for an optimal dimension of 20 - 25 nm, 10 - 15 nm. For the other molecules with an excitation spectrum longer than that from the SPR band of the Ag NP, a larger Ag core (e.g. ) can benefit the fluorescence by providing a broadband AEF to cover this excitation spectrum. In this paper, we have not considered the improvement of the intrinsic quantum yield of molecule by the reduction of fluorescence lifetime, caused by SPR. However, for a real measurement of AEF of coreshell, this factor could be also significant, thus a further study is needed. Moreover, AEF will also be useful for the study of the surface-enhanced fluorescence resonance energy transfer of two nearby molecules (donor and acceptor) [23,24] by using coreshells. In addition, the coherent light of a spaser-based nanolaser has been demonstrated recently by using molecules-doped Au@SiO2 structure , where our model of AEF could also be applied.
The research was supported by NSC, Taiwan, R.O.C. (NSC 98-2221-E-002-002, NSC 96-2221-E-182 021, NSC 97-2221-E-182-012-MY2).
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