Article pubs.acs.org/JPCC
Taking Plasmonic Core−Shell Nanoparticles Toward Laser Threshold Nils Calander, Dayong Jin, and E. M. Goldys* MQ BioFocus Research Centre, Macquarie University, North Ryde 2109 NSW, Australia S Supporting Information *
ABSTRACT: The first experimental demonstration of lasing plasmonic nanoparticles in 2009 ignited interest in active plasmonic structures with optical gain. However, the understanding of lasing in plasmonic nanoparticles is largely incomplete, and even less is known about their characteristics as they are taken toward the lasing threshold. Here we present a computational method and predictions of the lasing wavelength and threshold gain for spherical core−shell nanostructures with a metal core and a gain medium in the shell. We demonstrate that light scattering provides a simple diagnostics method to establish how far a specific nanoparticle is from reaching the lasing threshold. We also show that these structures can enhance the electric field by a factor of over 1500 (at 99.9% of threshold gain) and beyond, taking biosensing with these “smart dust” nanoparticles into the single molecule sensitivity regime.
1. INTRODUCTION High losses at visible frequencies in the metals hinder a number of interesting nanophotonics applications, but at the same time the metals provide important benefits such as exceptionally intense localization of electromagnetic fields. Hence the possibility of compensating these unavoidable losses by gain from an amplifying medium is actively pursued to enhance the desirable plassmonic effects without sacrificing performance.1−9 Nanoparticles with spherical core−shell geometry are a particularly attractive system for such studies. The development of wet chemistry techniques has now reached a stage where complicated core−shell designs can be carried out with ease.10−13 Investigations of structures with noble metal core and dielectric shell as well as the opposite structures with dielectric core and metal shell have been reported.14,15 Silicon dioxide is commonly used as a dielectric as its thickness can be precisely controlled and covalent binding of fluorophores can be carried out with high density. Theoretical studies of such composite plasmonic structures are still scarce. A study of metal nanoshells with a gain material in the core predicted the onset of laser action and discussed lasing characteristics.16 Another recent modeling effort17 predicted an increased Raman scattering cross-section in a complicated nanocage geometry not yet realized in practice. These encouraging predictions suggest that Raman and fluorescence sensing is likely to greatly benefit from gain-related effects especially that such nanoparticles can be used in a flexible format of “smart dust”.18 However, such applications require appropriate structure design, and this is addressed in the current paper. Our work focuses on spherical core−shell nanostructures with a metal core and a dielectric shell, the latter comprising a gain medium (Figure 1 a). In order to understand the lasing process in such composite plasmonic nanoparticles l we draw analogies with conventional laser cavities. The onset of lasing is possible only at those frequencies of electromagnetic field when © 2012 American Chemical Society
the structure experiences a resonance condition (eigenmode). A core−shell plasmonic nanoparticle with a metal core exhibits a number of eigenmodes (dipole and higher multipoles) of the electromagnetic field. These tend to merge due to large width of the multipole modes.19 The plasmonic nanoparticle considered here acts as an optical cavity localizing the optical wave around the metal nanostructure, although its Q factor tends to be limited (14.6 in ref 1). Furthermore, laser action is possible only at frequencies where the medium (small-signal) gain exceeds the loss coefficient. This condition selects the cavity mode where lasing is initiated with increasing gain. The first part of this paper establishes specific conditions for lasing in composite plasmonic nanoparticles. The formalism is based on exact analytical solutions of the Maxwell equations in spherical geometry by using the Mie theory.20,21 The calculations shown here were carried out for core−shell structures with the arrangement of layers shown in Figure 1a, and they can be easily extended to structures with an arbitrary number of concentric spherical shells. In our calculations, the inner sphere was made of a noble metal Au or Ag and the outer shell was a dielectric able to provide gain that is uniformly distributed across the structure. The dielectric function in the dielectric, ε2 = εr − iεi was assumed to have an additional negative imaginary term responsible for gain. The term εi is independent of wavelength; that is, we are assuming there is no dispersion. This is justified by the fact that the particle resonances are narrow compared to gain bandwidth, and completely within that bandwidth. Furthermore we use the same model formalism to discuss the light scattering characteristics of nanoparticles as they are taken toward the lasing threshold. We show that the scattering spectrum develops a pronounced spectral feature which tends Received: December 20, 2011 Revised: February 26, 2012 Published: March 9, 2012 7546
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shown to be an effective way to amplify fluorescence or Raman signals from molecules conjugated to the nanolaser surface.
2. RESULTS AND DISCUSSION 2.1. Calculation of Lasing Frequency and Threshold Gain. In order to calculate the threshold gain for lasing we consider the characteristics of the eigenmodes of the structure in the presence of gain. To this aim we evaluate the resonance frequencies of the dipolar and higher modes by finding nontrivial solution of the Maxwell’s equations that is finite at the center of the core. Additionally, far from the center, this solution contains only the outgoing wave that tends to zero at infinity (see the Supporting Information for more details). The dielectric function in the metal (gold or silver) was taken from ref 22. The lasing frequency is determined by solving a system of equations derived from matching the tangential components of the electric and magnetic fields at the boundaries of the inner and outer surface of the spherical shell (Equation S1 in the Supporting Information), in a way similar as in refs 23 and 24. This matching is possible only for selected values of the wave vector of the spherical wave, which specifies the threshold lasing condition for the real and imaginary part of a dielectric function, at each frequency. The value of the refractive index of a dielectric at zero gain determines the eigenmode frequency where lasing is possible for a given structure geometry. This is illustrated in Figure 1b where we show the refractive index of the dielectric at zero gain, εr1/2, for different eigenmodes and the corresponding values of imaginary part of the refractive index k = ±2−1/2(−εr + (εi2 + εr2)1/2)1/2 as a function of lasing wavelength. The latter is related to the optical gain coefficient by g = 4πk/λ. Our calculations shown in Figure 1 b have been carried out for the nanoparticle geometry identical to that reported by Noginov.1 The lasing is taking place in a dipolar mode and we note that our predictions of the lasing wavelength for this geometry (527.3 nm) corresponds closely to 531 nm measured in Noginov’s experiment. 2.2. Example Numerical Results for Various Au and Ag Core Structures. Supported with this experimental validation of our model we have extended our simulations to other similar geometries within the experimentally achievable range of sizes. In these simulations we used both gold and silver while the refractive index of the gain medium was taken as 1.47, consistent with the average refractive index of silica.25 The geometries discussed here are listed in Table 1. The first geometry with a 14 nm diameter core (d = 14 nm) and a silica shell with thickness t = 15 nm is identical to that shown to lase in ref 1. The value of threshold gain for this structure is predicted to be 9.77 × 104 cm−1. The conditions for lasing are somewhat more favorable in structures with a large
Figure 1. (a) Core−shell geometry of the nanostructures. (b−e) results obtained for the d = 14, t = 15 nm Au core structure as in ref 1. (b) The real part of the refractive index at zero gain, (εr)1/2, and the imaginary part of the refractive index in the presence of gain, k, as a function of wavelength for the lowest few eigenmodes. The horizontal line at the refractive index of silica (n = 1.47) identifies the mode frequency. The mode with the smallest |k| is the first lasing mode. (c and d) magnified view of the panel b showing the frequencies of the eigenmodes and the corresponding values of k for different modes. (e and f) the calculated patterns of local field intensity in the lowest modes, (a) dipolar and (b) quadrupolar.
to the lasing wavelength and grows as the gain in the structure approaches the lasing threshold. Such measurements of scattering spectra as a function of wavelength at high pumping power, or scattering intensity at the lasing wavelength as a function of excitation power offer a simple diagnostics method to assess whether a nanoparticle is capable of lasing and how far it is from the lasing threshold. Finally, we evaluate the electric field at the surface of such structures on approach to lasing, which is also significantly enhanced compared with the same structures at zero gain. Thus pumping the gain medium in a nanolaser is
Table 1. Details of the Examined Nanostructure Geometries, All with SiO2 Shells, Lasing Characteristics, and Enhancement of the Electromagnetic Field at the Lasing Wavelength Averaged over the Nanoparticle Surface core material core diameter, d = 2R1 (nm), shell thickness t = R2 − R1 (nm) lasing mode lasing wavelength (nm) imaginary part of the refractive index klas at threshold threshold gain coefficient g = 4πklas/λlas (cm−1) average electric field enhancement factor at lasing wavelength at 99.9% gain on the outside shell 7547
Au 14 15 dipolar 527.3 0.41 9.77 × 104 160
Au 50 50 dipolar 542.8 0.30 6.97 × 104 430
Ag 14 15 dipolar 399.9 0.038 1.19 × 104 1500
Ag 30 15 quadru-polar 378.7 0.043 1.43 × 104 320
Ag 50 50 quadrupolar 383.3 0.038 1.25 × 104 170
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core diameter and a thicker shell (d = 50 nm and t = 50 nm, respectively), where the threshold gain of 6.97 × 104 cm−1 is found. Here the lasing mode spreads out over a thicker shell and its smaller fraction extends over the lossy core region compared to the previous geometry (d = 14 nm, t = 15 nm). For this large structure the lasing wavelength is also comparatively longer (542.8 nm). This longer lasing wavelength is further away from the energy-dissipating interband transitions in the metal, hence favorable influence on the laser threshold. The inspection of the nanoparticles with Ag core leads to somewhat different observations. The structure with d = 14 nm, t = 15 nm is lasing in a dipolar mode, at a much shorter wavelength of 399.9 nm than the corresponding Au core structure, with a very small threshold gain of 1.19 × 104 cm−1. This is due to smaller losses in Ag than in Au. However even a small increase of the core diameter, to 30 nm, changes the lasing mode from dipole into the quadrupole, with further decrease of the lasing wavelength to 378.7 nm, and increased threshold gain to 1.43 × 104 cm−1 (Figure 2). This mode switch
378.8 nm. Although the number of fluorescent dyes emitting between 370 to 400 nm is limited and they are relatively weak comparing with the visible dyes, their emission cross sections are sufficient. For example, Methoxycoumarin dyes (succinimidyl ester with excitation peak at ∼350 nm; emission peak ∼380 nm; Invitrogen), capable of binding to APS and compatible with the current covalent-binding silica nanoparticle chemistry have absorption cross section of in excess of 1.6 × 10−16 cm2.26,27 Lasing in Nanoparticles with Other Types of Shell Material. It is interesting to explore whether lasing is possible for gain materials other than silica. Our calculations confirm this, and provide lasing conditions, as well as the lasing frequency. In order to find the lasing wavelength for a structure with a shell made from a given material, one should plot the real part of the refractive index at zero gain, εr1/2, and the imaginary part of the refractive index in the presence of gain, k, as a function of wavelength (λ) for the lowest few eigenmodes. The intersection of a horizontal line at the refractive index of the material under consideration with the graph of εr1/2(λ) gives the lasing mode frequency λlas. The value of klas = k(λlas) identifies the lasing threshold and the threshold gain coefficient is calculated as g = 4πklas/λlas. The mode with the smallest |klas| is the first lasing mode. For arbitrary geometries other than the geometries discussed earlier numerical calculations are required, by using Equation S1 in the Supporting Information. For the geometries listed in Table 1, it is sufficient to consult Figures 1, 2, S1, S2, and S3. These figures illustrate that the nanoparticles with a gain medium with higher refractive index than silica are capable of lasing at longer wavelengths than silica−based core− shell nanostructures, and that the threshold k for such structures becomes, generally, lower than for silica. Materials with lower refractive indices than silica are capable of lasing as well, at shorter wavelengths than the corresponding silica structures. Hence our results can be used to inform the design of lasing nanoparticles based on a broad range of gain media. 2.3. Experimental Diagnostics to Verify the Proximity of Lasing in the Nanoparticles. Further we discuss how to assess the proximity of a given nanoparticle from the lasing threshold in specific experimental conditions. Conventional methods used for laser media are not applicable for nanoparticles where the gain medium and the cavity are inseparable. Our method is based on relatively uncomplicated measurements of the 90° scattering cross section of diluted nanoparticle suspensions as a function of wavelength. This scattering cross section need to be measured with and without gain. Our calculations (Figures 3a and 4) indicate that such measurements can accurately indicate how far the nanoparticles are from the lasing threshold. As an example, we discuss the scattering cross section at 90°, measured at the lasing wavelength (Figure 3a). These calculations indicate that as gain of a lasing nanoparticle is increased, the scattering cross-section at the lasing wavelength rapidly increases. This quantity can be measured in a conventional scattering experiment with monochromatic, variable or even single wavelength excitation. In the latter case the measurements should be done at the lasing wavelength which can be calculated by using the approach presented earlier. A second beam needs to be provided separately, this beam optically pumps the nanoparticles realizing gain. Two measurements of the scattering cross-section need to be carried out, with the pump on and off. The ratio of these two signals provides the information on the magnitude of gain in the examined nanoparticles in relation to the lasing threshold gain.
Figure 2. Results obtained for the d = 30 nm t = 15 nm Ag core structure lasing in a quadrupolar mode (a). The real part of the refractive index at zero gain, εr1/2, and the imaginary part of the refractive index in the presence of gain, k, as a function of wavelength for the lowest few eigenmodes. The horizontal line at the refractive index of silica (n = 1.47) identifies the mode frequency. The lasing mode is one with the smallest |k|. (b and c) Magnified view of panel a showing the frequencies of the eigenmodes and the corresponding values of k for different modes.
is induced by the fact that in the quadrupole mode the electric field is concentrated rather in the outer part of the metal core, which lowers the losses. Upon further increase of core and shell size, to d = 50 nm, t = 50 nm the lasing continues in the quadrupole mode and the lasing wavelength increases to 383.3 nm while the threshold gain decreases somewhat, to 1.25 × 104 cm−1 (see the Supporting Information, Figures S1−S3 for the results in the remaining nanoparticles). Our results show that all the examined nanoparticles are capable of lasing, provided the threshold gain g can be achieved by using suitable fluorophores. Silver core appears especially favorable for lasing as, due to smaller losses in Ag than Au, the threshold gain for silver is about 1 order of magnitude lower than in gold with values between 1.19 × 104cm−1 and 1.43 × 104cm−1 (Table 1). The gain coefficient is a product of concentration N and the emission cross section σem of gain molecules, g = Nσem. With the current progress in chemistry of covalent-binding of fluorophores into silica nanoparticles chemistry the concentrations in the order of ∼6.25 × 1019 cm−3 have been achieved.1 To reach the lasing threshold the UV emitting dyes are required with an emission crosssection of 2.28 × 10−16 cm2 at 399.9 nm or 1.91 × 10−16 cm2 at 7548
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Figure 3. (a) Scattering cross section at 90° at the lasing wavelength for the examined structures. (b) Average electric field at the outer surface of the nanostructures (relative to the amplitude of the incident wave) close to 99.9% of the gain threshold. Colors indicate different geometries as shown in the Figure, asterisk indicates nanoparticles where lasing is initiated in a quadrupolar mode.
empirical value of the lasing threshold we can now return to Figure 3 to estimate how far is the examined nanoparticle from reaching the lasing threshold. 2.4. Electric Field Enhancement at Nanostructure Surface As a Result of Gain. Finally we discuss the electric field enhancement in these structures. Plasmonic nanoparticles of various kinds have long been known to cause the enhancement of electric field in their proximity. This quantity is significant for biosensing applications, in particular for Raman and fluorescence sensing, that benefit from enhanced average electric field at the surface of the nanoparticles. In such application, a biosensing event, that is immobilization of an analyte molecule at the nanoparticle surface is detected either as its Raman signature or fluorescence signal from a fluorescently labeled analyte. In both cases the enhanced electric field increases the signal. It is interesting to explore this effect in nanoparticles with gain as they are excited just below the lasing threshold. In Figure 3b we have plotted the average electric field at the lasing wavelength at the outside of the nanoparticle shell (in water). The curve increases steeply with increasing gain, indicating that the presence of gain enhances the electric field compared to the situation on the surface of the same nanoparticle without gain. The enhancement is calculated relative to the amplitude of the incident wave. The increase is very significant, and factors in the order of over 1000 are easily achieved at 99.9% gain. Table 1 summarizes the electric field enhancement values at the surface of the nanoparticles obtained at high gain (99.9% of lasing threshold) at the lasing wavelength for all geometries examined here. We note that Ag-based structure lasing in a dipolar mode (d = 14 nm, t = 15 nm) shows the highest field enhancement of 1500. The corresponding Au-based structure enhances field less efficiently, by a factor of 160. The large Au structure (d = 50 nm, t = 50 nm) shows a larger enhancement factor of 430. The Ag-based structures lasing in a quadrupole mode are less effective at enhancing the field, however still a factor of 320 is obtained for the d = 30 nm, t = 15 nm structure.
Figure 4. Scattering cross section at 90° as a function of wavelength for representative nanostructures. (a) Au core, d = 14 nm, t = 15 nm; (b) Au core, d = 50 nm, t = 50 nm; (c) Ag core d = 14 nm, t = 15 nm; (d) Ag core, d = 50 nm, t = 50 nm, the latter structure starts to lase in a quadrupolar mode. Colors indicate % threshold gain. Red, 100%; green, 50%; blue, 20%; black, 0%.
Thus it provides information how far a given nanoparticle is from the lasing threshold. Without prior knowledge of the lasing wavelength, spectral dependence of the scattering crosssection needs to be measured, with and without gain. In Figures 4 and S4, we show such wavelength dependence of the scattering cross section (at 90°) in the examined structures. The two quantities have been calculated for varying levels of nanoparticle gain, assumed to be constant across the entire examined wavelength range. We note that the scattering cross-section evolves as gain is increased. It has a characteristic Fano line shape, especially well pronounced in gold-based nanostructures (Figure 4a,b). The Fano resonance occurs from interference between resonance modes, as explained in more detail (without gain) in ref 28. We also note that the Fano effect is exaggerated by gain, which, to the best of our knowledge, has not been reported previously. Close to the lasing threshold the scattering cross section develops a very sharp peak that, at the threshold gain, coincides with the lasing wavelength (similar effects are also present in the extinction cross-section). Thus the scattering cross section provides the information about lasing wavelength which becomes more accurate as the threshold is approached. With the
3. CONCLUSIONS In this publication we described how to calculate the lasing frequency of composite core−shell noble metal-dielectric nanoparticles. We examined several core shell geometries with varying sizes and established trends for the lasing frequency and threshold gain. Furthermore we suggested a method based on scattering measurements to monitor progress of a nanoparticle toward the onset of lasing. This method can also help estimate the lasing wavelength without numerical simulations, even though the Fano effect affects the location of 7549
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carried out in similarly as in ref 21 (see the Supporting Information for more details). The electromagnetic field in each of the regions is expressed in terms of spherical multipole (Mie) functions of the lth order (l = 1 for dipole resonance, l = 2 for quadrupole, l = 3 octupole etc). For the geometries considered here we only need to include electric multipole functions because only such modes can be excited in a structure with radius that is smaller than the light wavelength. Since the field has to be finite in all points inside the core, in this region we use an electric multiple function based on a spherical Bessel (j) function of the first kind, Nl,0 (k1, r, θ, φ), where l is the orbital momentum quantum number. Here k1,2,3 = ω(ε1,2,3)1/2/c and c is the speed of light. Inside the shell layer we use a linear combination of the multipole functions based on spherical (h1) Hankel functions of the first and second kind, Nl,0 (k2, r, θ, φ) (h2) and Nl,0 (k2, r, θ, φ), respectively. The solution in the outside medium is sought as a multipole function based on a spherical (h1) Hankel function Nl,0 (k3, r, θ, φ) because it does not diverge at infinity, while its counterpart does. We impose the usual continuity conditions for the components of the electric and magnetic fields at internal and external surfaces of the shell. Enforcing the condition that we have only an outgoing wave in the medium determines the wave vector k3 for each l and hence the wavelength of the eigenmode, λl0. These frequencies are degenerate with respect to the quantum number m due to spherical symmetry, hence our choice of m = 0. Our analytical calculations have been confirmed by numerical solution of the Maxwell’s equations by using COMSOL software.
the spectral features. Finally we have quantified the enhancement of electric field on the nanoparticle surface that is fuelled by the gain in the medium. The enhancement factors for the structures discussed here are over 3 orders of magnitude higher the enhancement observed in plasmonic nanoparticles without gain. Such enhancement factor brings the sensitivity of Raman effect into the single molecule regime. Our work can help pave the way for further development of broad classes of lasing plasmonic nanoparticles. These can be constructed from silica by using covalently bound dyes as in ref 1. Here we provide conditions for dye selection, such as emission wavelength and excitation cross section required to reach threshold gain, as well as the required dye concentration. We also show feasibility of lasing in Ag core particles, where ohmic losses are smaller than in Au. We demonstrate that alternative materials can also be used for nanolasers, and our work provides information on the lasing wavelength for a given refractive index which is required to match optical gain of the material to cavity properties. Again, information on threshold gain is provided. The experimental evaluation of the nanolasing effect typically involve measurement of line narrowing and threshold behavior of the observed signal;1 however, these effects happen after the threshold has been reached. Here we describe the optical signatures of nanolasers as they have not yet reached the laser threshold, which could be more useful than the conventional criterium of line narrowing. As a final comment we bring to the attention of the relevant research community that demonstration of lasing in nanoparticles needs to be experimentally distinguished from random lasing,29 preferably in single particle studies, or by using a dilution sequence of the nanoparticle solution. This approach is needed to distinguish random lasing within a collection of particles where lasing occurs due to random optical scattering loops, from lasing of a true single nanoparticle laser.
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ASSOCIATED CONTENT
S Supporting Information *
Analytical model to compute lasing wavelengths. This information is available free of charge via the Internet at http://pubs.acs.org
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4. METHODS We evaluated the resonance frequencies of the dipolar and higher modes by finding nontrivial solution of the Maxwell’s equations that is finite at the center of the core and tends to zero at infinity (for more details see the section “Analytical model to compute lasing wavelengths” in the Supporting Information) The lasing frequency is determined by solving a system of equations derived from matching the tangential components of the electric and magnetic fields at the boundaries of the inner and outer surface of the spherical shell (Equation S1, Supporting Information), in a way similar as in refs 23 and 24. This matching is possible only for selected values of the wave vector of the spherical wave, which specifies the condition for real and imaginary part of a dielectric function, for each frequency. The proper choice of the dielectric function is crucial for predicting the resonance characteristics of nanoparticles in experimental situations. In the calculations shown here we used the bulk dielectric function in the metal from the work by Johnson and Christy22 which ignores the effect of “mean free path” correction applicable to nanoscale metals. However Nordlander demonstrated that these corrections are not necessary to accurately model gold nanoparticles, and that the Johnson and Christie dielectric function is adequate.30 Nonlocal effects have also been ignored, as they begin to become pronounced only at a single nanometre scale.31 The calculation of the eigenmodes of a structure made of a core surrounded by a concentric shell placed in the medium is
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS Discussions with D. Coutts, J. Dawes, and M. Steel are gratefully acknowledged. REFERENCES
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