Introduction

Plasmonic nanolasers are different from conventional lasers in that surface plasmons [1–3], rather than photons, amplifying the radiation by stimulated emission. A plasmonic nanolaser is a quantum generator and can generate strongly localized surface plasmon resonances on the nanoscale [4, 5]. Its physical size is much smaller than a conventional laser, which helps applications for medical technology, sensing, and information storage. However, plasmonic nanolasers with large lasing thresholds because of high dissipative loss are undesirable for practical applications, and the choice of gain materials is limited to obtain useful thresholds [6–8]. Many researchers have constructed different plasmonic nanolaser structures to improve performance, such as nanowire on silver substrate [9], single nanoparticle covered by gain material [10], and graphene-embedded hybrid plasmonic resonator [11], the crescent nanostructure with gain medium inside [12] and Ag nanoparticles with various shapes [13]. Among them, and many other works, it has shown that a high-quality factor is often associated with a low threshold plasmonic nanolaser operation [9, 10, 12–18].

Ugo Fano first established the existence of resonances with asymmetric line shapes with the theory of configuration in 1961 [19–21]. Fano resonances are highly sensitive to the surrounding environment, which can lead to strong localized electric field enhancement and quality factor [22–24]. Thus, a plasmonic nanolaser based on Fano resonances should produce a low threshold, and two plasmonic nanolaser types based on Fano resonances have been proposed [25, 26]. Although low thresholds have been obtained from their systems, both systems were designed or fabricated based on planar integrated configuration, and their characteristics are highly dependent on the precision control of the shape and position of each nanostructure in the gap system. In addition, the planar integrated configuration of optical gap systems is not flexible enough for photovoltaic and biological applications. Therefore, the development of metallic nanostructures with both significant near-field enhancement, perhaps comparable to that of a typical gap system, and flexible nanoparticle (NP) morphology, for achieving a low threshold, remains essential to actual applications.

In the present paper, a novel plasmonic nanolaser design will be demonstrated using a heptamer system among silver (Ag) nanoparticles. Similar heptamer system has widely been fabricated for different applications [27–30]. As a numerical sample, a heptamer covered with gain material will be designed and characterized, which has a super low threshold. The heptamer scattering spectrum exhibits narrow Fano resonance, caused by interference between nanoparticle’s bright and dark modes [27, 31]. Many important optical properties (such as the localized electric field intensity, Fano resonance wavelength, and optical cross section) were analyzed using the finite element method (FEM). Compared with the planar integrated configuration of optical gap systems [25, 26], the present NP-based laser design has more flexible structure so that it is well suited for various applications.

Simulation Model and Method

We start our analysis by calculating the electric field intensity and optical cross section of the heptamer surrounded by gain material. The simulation setup is shown in Fig. 1. Perfectly matched layers (PMLs) were used to eliminate redundant scattered waves at the boundaries for all directions. A PML 150 nm thick was sufficient to remove the effect from all non-physical scattering in the analyzed simulations. The simulation domain was a cube box, with 800-nm sides. The entire region was meshed at 10 nm, with a smaller mesh grid (0.2 nm) used for the nanoparticles. Optical constants for silver were described using the Johnson and Christy model [32]. Typically, the gain material around the heptamer would be silica-doped semiconductor-doped or with optical gain inclusions, e.g., organic dyes or rare earth ions.

Fig. 1
figure 1

Heptamer of silver nanoparticles and spherical gain material. a Simulation stereogram, b cross section view in the x-y plane

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For the proposed theoretical model, the refractive index of the gain material was represented by n g. The gain coefficient was assumed to be independent of frequency and was introduced through the imaginary part of the dielectric permittivity, \( k=Im\sqrt{\varepsilon^{\prime }+i{\varepsilon}^{{\prime\prime} }} \). Therefore, the complex refractive index of the gain material is n g −i × k, where a negative imaginary part means an active gain. The gain material radius was fixed to 200 nm for the simulations. Outside of the gain material is air with refractive index n Air. Using COMSOL Multiphysics 5.1 software, the localized electric field can be shown to be greatly enhanced with a super low gain coefficient at the Fano resonance wavelength. Note that the localized electric field can be greatly enhanced with a suitable gain coefficient, which is the plasmonic nanolaser threshold.

Numerical Analysis and Results

Heptamer Optical Properties

Incident light is linearly polarized along the x-axis with incident field intensity E 0 = 1 V/m and spreads along the direction of negative z-axis. The heptamer is surrounded by a spherical dielectric material. To understand the heptamer optical properties, let us first analyze the electric field and optical cross section, where (see Fig. 1b) the Ag nanoparticle radius is r = 50 nm; the dielectric material radius is R = 200 nm; the gap distance is d = 5 nm; the refractive index is n g = 1.5, similar to typical glassy material; and the gain coefficient is k = 0, i.e., no gain medium in the dielectric material. Figure 2 shows the spectrum of the normalized electric field, |E|/|E 0 | and the optical cross section in the dielectric material. The gain coefficient is dispersed very little around the resonant peak and hardly influences the threshold [33–35]. Therefore, the dispersion of the gain coefficient was ignored, without loss of generality, to simplify the analysis. Thus, the gain coefficients were dispersionless in the optical spectra of absorption, scattering, and extinction cross sections.

Fig. 2
figure 2

Electrical and optical properties of the proposed plasmonic nanolaser. a Normalized electric field |E|/|E 0 | spectrum of the heptamer. Inset: electric field distributions at three wave peaks marked as a, b, and c. b Optical cross section spectra of absorption, scattering, and extinction of the heptamer. c Electric field distribution and maximum electric field intensity the four points marked as A, B, C, and D in (b)

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The maximum electric field intensity position is not fixed in the simulation model and may change with the strength of interaction among the Ag nanoparticles [33]. Therefore, we used the highest electric field (E) along the x-y plane to indicate the strength of the interaction. As shown in Fig. 2a, there are two obvious wave peaks in the visible and one in the near-infrared ranges, at wavelengths 428, 500, and 810 nm, respectively. The electric field distributions, the three wave peaks, marked as a, b, and c in Fig. 2a, are analyzed in the x-y plane, shown in the inset figures.

The optical properties of symmetric plasmonic heptamers composed of spherical and cylindrical nanoparticles have been analyzed and described in several publications [31, 36–38]. The two relevant modes for Fano interference are a bonding bright (superradiant) mode where the dipolar plasmons of all nanoparticles oscillate in phase and in the same direction, and an antibonding dark (subradiant) mode, where the dipolar moment of the center particle opposes the dipole moment of the surrounding ring. In the retarded limit, the bright mode becomes superradiant, while the dark mode remains subradiant, and the weak coupling mediated by the plasmonic near-field introduces an interaction between the subradiant and superradiant modes, inducing a Fano resonance in the superradiant continuum at the energy of the subradiant mode. Figure 2b shows the Fano resonance appears at 828 nm. Four points in the scattering cross section spectrum, marked as A, B, C, and D, were used to compare maximum electric field intensity. Point C, at the Fano resonance wavelength, has the highest electric field intensity (106 V/m) (see Fig. 2c).

Effect of the Dielectric Material Gain Coefficient on Laser Properties

The imaginary part of the complex refractive index represents the dielectric material gain coefficient. A suitable gain coefficient can greatly enhanced localized electric field intensity. When the gain coefficient = 0.06, the normalized electric field |E|/|E 0 | and optical cross section were significantly changed and enhanced at the Fano resonance wavelength (λ = 823 nm) (see Fig. 3), and there is only a single peak in the normalized electric field spectrum (Fig. 3a). The scattering cross section and electric field intensity at point B (the Fano resonance wavelength) are higher than at points A and C. Indeed, the maximum electric field intensity at Fano resonance is the highest of the whole spectrum.

Fig. 3
figure 3

Electrical and optical properties of the proposed plasmonic nanolaser with gain coefficient (k = 0.06). a Normalized electric field |E|/|E 0 | and extinction cross section. b Absorption and scattering cross sections, and insets show with electric field distribution and maximum electric field intensity at the marked points

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However, this gain coefficient (k = 0.06) is not necessarily the plasmonic nanolaser threshold. The maximum value of the scattering cross section defines the plasmonic nanolaser threshold as the gain coefficient increases from zero. Figure 4a shows that the optical cross section and normalized electric field intensity are sharply peaked at gain coefficient = 0.058, i.e., the plasmonic nanolaser threshold = 0.058 for the considered structure parameters. When the gain coefficient is less than the threshold, a better compensation of energy loss can be obtained as the gain coefficient increases. When the gain coefficient is slightly larger than the threshold, the laser phenomenon occurs. However, as gain coefficient further increases, the localized electric field intensity will decrease among the nanoparticles.

Fig. 4
figure 4

a Optical cross section at the Fano resonance wavelength as gain coefficient increases. Inset: electric field distribution and maximum electric field intensity at point A (λ = 823.2 nm, k = 0.058). b Far-field distribution at various radiation angles in the x-y plane with five gain coefficients for incident wavelength = 823.2 nm

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The direction of excitation light is an important laser parameter, and the polar plot is very useful to evaluate the light emitting equipment performance. Figure 4b shows the plasmonic nanolaser polar plot, with normalized far-field intensities and distributions for gain coefficient from 0.055 to 0.059. When gain coefficient equals to 0.058 (i.e., at the threshold point), the maximum normalized far-field intensity is around 0.0002. The far-field electric field performs a dipole, and the direction of excitation light is perpendicular to the incident direction and polarization direction of incident light, simultaneously. Therefore, the proposed structure emission has very good directivity. The optical cross section from Figs. 3 and 4 show that the gain coefficient has a large influence on high energy light production.

Effect of Dielectric Material Refractive Index on Laser Properties

The choice of dielectric material is very important to the plasmonic nanolaser development. Figure 5a shows that increasing the dielectric refractive index over a small range (1 < n g < 2) shifts the Fano resonance toward longer wavelengths, but the scattering cross section remains the same. In addition, both the Fano resonance wavelength and laser threshold increase linearly with refractive index increase (see Fig. 5b). From Fig. 5b, one also sees that the threshold always keeps a low value in the whole calculated range of the refractive index.

Fig. 5
figure 5

Optical and electrical properties change with changing refractive index (1 < n g < 2). a Scattering cross section, Fano resonances are marked as A, B, C, D, and E. b Fano resonance wavelength and laser threshold. Inset: electric field distribution and the maximum electric field intensity at 715 nm (n g = 1.3, k = 0.045)

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The refractive indexes of many dielectric materials are higher than two, e.g., semiconductors, and Fig. 6a shows change trend of Fano resonance wavelength with higher refractive indexes. The Fano resonance is redshifted as refractive index increases from 1.5 to 3.5 and the scattering cross section remains the same on the Fano resonance. However, the scattering cross section spectra gradually show asymmetry and wider bandwidth as refractive index increases, particularly on the left side. Figure 6b shows the relationship between laser threshold and refractive index. Both dielectric material refractive index and nanoparticle size go together with the value of scattering cross section for the heptamer system. For the present nanoparticle size, the maximum scattering cross section occurred at refractive index = 2. Thus, a maximum gain was also needed to compensate the scattering. Therefore, the highest threshold occurs at refractive index = 2, and the threshold reduces as the refractive index increases further. The result suggests that a relatively low or a relatively high refractive index of dielectric material can be used with a low threshold. Similarly, the Fano resonance wavelength linearly increases with increased refractive index.

Fig. 6
figure 6

Optical and electrical properties change with large changes in refractive index. a Scattering cross section with refractive index (1 < n g < 4), the Fano resonances are marked as A to E. b Fano resonance wavelength and laser threshold as refractive index increases. Inset: electric field distribution and maximum electric field intensity at 1105 nm (n g = 2.0, k = 0.082)

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Effect of Gap Distance between Adjacent Ag Nanoparticles on Laser Properties

The gap distance between adjacent Ag nanoparticles is crucial for determining the interaction between them. There are some approaches to control the gap distance among nanoparticles, such as the use of molecular linkers, asymmetric functionalization of the linkers to the NP building blocks, and the control of the aggregation kinetics by destabilizing the electrostatic interactions of a colloidal system [39]. The synthesis of silver nanoparticles can be performed using the Turkevich method, which is based on the reduction properties of boiling citrate solutions [40]. According to this approach, it can achieve the formation of close-packed aggregates (including the heptamer of silver nanoparticles) with a gap distance of around 5 nm. Therefore, the gap distance was varied from 3 to 7 nm to investigate the effects on plasmonic nanolaser optical properties, as shown in Fig. 7. Figure 7a shows that the Fano resonance shifts toward shorter wavelengths as the gap increases, while the scattering cross section linearly increases. The intensity of Fano resonance is closely related to the gap distance between adjacent Ag nanoparticles. For the present numerical example, an optimal Fano resonance occurred when the gap distance equals to 4 nm. Therefore, Fig. 7b shows there is an optimal gap distance to generate the lowest threshold, and we may select an appropriate gap distance to tune plasmonic nanolaser performance. That is to say an appropriate gap distance can be used with the lowest threshold for a given radius of nanoparticles.

Fig. 7
figure 7

Optical and electrical properties change with changing nanoparticle gap distance. a Scattering cross section, Fano resonances are marked as A to E. b Laser threshold and Fano resonance wavelength. Inset: electric field distribution and maximum electric field intensity at 842.6 nm (d = 4 nm, k = 0.048)

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Effect of Ag Nanoparticle Radius on Laser Properties

When the gap distance between adjacent nanoparticles remains constant, the nanoparticle size determines the plasmonic nanolaser size. Figure 8 shows scattering cross sections, plasmonic nanolaser threshold, and the Fano resonance for different Ag nanoparticle radii. Fano resonances exhibit a redshift as the radius increases (Fig. 8a), which indicates a larger plasmonic nanolaser will excite light with longer wavelengths. At the same time, the scattering cross section linearly decreases as nanoparticle radius increases. However, when nanoparticle radius changes from 40 to 60 nm, plasmonic nanolaser threshold remains almost constant. This is because both scattering cross section and gain (i.e., negative absorption) cross section at Fano resonance reduce as the radius increases in equal proportion.

Fig. 8
figure 8

Optical and electrical properties change with changing nanoparticle radius. a Scattering cross section, Fano resonances are marked as A to E. b Laser threshold and Fano resonance. Inset: electric field distribution and maximum electric field intensity at 767.1 nm (r = 45 nm, k = 0.059)

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Effect of Central Ag Nanoparticle Radius on Laser Properties

The nanoparticle at the center of the heptamer interacts with six circumjacent nanoparticles and plays a very important role in determining plasmonic nanolaser properties. Therefore, the influences of changing this radius on Fano resonance wavelength and laser threshold were investigated. Figure 9 shows the scattering cross sections, plasmonic nanolaser thresholds, and Fano resonances for different central nanoparticle radii. Fano resonance shifts toward shorter wavelength and exhibit narrower full width half maximum as the radius decreases (Fig. 9a). Fano resonance linearly shifts to longer wavelengths as the radius increases, but the threshold shows a more quadratic response because the mode is localized around the central nanoparticle and then scales with the surface of the sphere (r 2). The plasmonic nanolaser threshold reduces to 0.014 at radius = 45 nm with localized electric field = 2.83 × 104 V/m. The laser threshold continues to reduce as the radius decreases, but the localized electric field intensity significantly reduces because of the weak interaction. Thus, 45 nm was the smallest central nanoparticle radius analyzed.

Fig. 9
figure 9

Optical and electrical properties change with changing central nanoparticle radius. a Scattering cross section, Fano resonance is marked as A to H. b Laser threshold and Fano resonance. Inset: electric field distribution and maximum electric field intensity at 746 nm (r center = 45 nm, k = 0.014)

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Conclusion

A novel plasmonic nanolaser was proposed by placing seven identical silver nanoparticles in a dielectric material to achieve significant localized electric field enhancement and reduced plasmonic nanolaser threshold. Simulations showed the effect of various parameters on plasmonic nanolaser optical and electrical properties. A heptamer immersed in dielectric material can produce Fano resonances and lower thresholds. The localized electric field among Ag nanoparticles can extremely be amplified as the gain coefficient reaches to the threshold. The Fano resonance wavelength will red or blueshift for different dielectric material refractive indexes, gap distances, and Ag nanoparticle radii. Selecting appropriate parameters can provide enhanced localized electric field beyond 104 V/m and reduced plasmonic nanolaser threshold of 0.014 at the Fano resonance wavelength.