Introduction

A collective resonant response to stimulating electromagnetic fields by the conduction electrons in metallic nanoparticles is widely known as localized surface plasmon resonance (LSPR), whose characteristic peak depends on geometrical and compositional factors, as well as on the nature of the incident light (i.e., polarization, intensity and wavelength) [1–6].

While for bulky nanoparticles (tens of nanometers or more in typical length), behavior of the LSPR is well understood in terms of the classical Drude’s model, recently reported blue-shifts of the LSPR in particles with radius below 10 nm, reveals the need for a quantum treatment of the involved carriers in order to more accurately describe the collective response in such strongly confined systems [7–13].

Understanding of the crossing between classical and quantum regimes and its effects on the optical response in this kind of systems, in addition to be appealing since the fundamental point of view, is becoming more and more relevant because of growing interest in nanoplasmonics applications.

Plasmonic resonances in isolated nanoparticles under polarized electromagnetic radiation, produce a strong scattered electric field as compared to the intensity of the incident field [14–17]. This effect is promising for technological applications given that augmented fields open doors to more efficient scenarios for strong radiation-matter coupling in low-dimensional semiconductor structures [18–21], and for magnified non-linearities in polar materials [22–24].

In this work, we study such an enhancement effect by using classical and quantum models in obtaining the dielectric functions. This allows us to clearly compare the optical response from both approaches.

The paper is organized as follows: In the first part the electromagnetic problem is described in terms of the dielectric properties of the nanoparticle. In the second part, those dielectric functions are calculated in the framework of the classical Drude’s model and also within a model that includes electron energy discretization from quantum confinement. In the last part, the near enhanced fields obtained from both approaches are compared, and a summary and conclusions are provided.

Field Enhancement Factor

The electric field scattered by the nanosphere must satisfy the Maxwell equations in matter. This leads to the Helmholtz wave equation for the electric field vector E, which in terms of the complex relative permitivity 𝜖 r , reads

$$ \nabla \times \mu_{r}\left( \nabla \times \mathbf{E}\right)-{k_{0}^{2}}\left( \epsilon_{r} - \frac{i\sigma}{\omega\epsilon_{0}}\right)\mathbf{E}=0, $$
(1)

where μ r is the relative permeability (taken as 1 for metallic systems), σ the nanoparticle conductivity, and \(k_{0}=\omega \sqrt {\epsilon _{0}\mu _{0}}=\frac {\omega }{c_{0}}\) (with c 0, 𝜖 0 and μ 0 the light velocity, vacuum permeability, and vacuum permitivity, respectively) is the incident light’s wave number.

The dielectric function is required as input parameter to solve Eq. 1 , and that function in turn depends on the frequency of the incident electromagnetic wave 𝜖 r (ω) = 𝜖 1(ω) + i 𝜖 2(ω).

Once this input function is established, the vectorial wave Eq. 1 can in principle be solved, yet in most cases numerical treatment is necessary [25]. In this case, we solve computationally this complex equation by means of a standard finite element method [26–30] (http://www.comsol.com).

We define the field enhancement factor (FEF) as the squared norm of the ratio between the scattered electric field E o u t in any point of the nanosphere surrounding (near field), and the amplitude of the incident z-polarized electric field E i n c

$$ \text{FEF}=\frac{|\mathbf{E}_{out}|^{2}}{|\mathbf{E}_{inc}|^{2}}. $$
(2)

The studied system is represented in Fig. 1, where the polarization effect of the incident wave on the conduction electronic plasma in the nanosphere, and the associated electric field modification are depicted.

Fig. 1
figure 1

Schematics of the stimulated plasmonic polarization in the nanosphere and the corresponding electric field modification in its surrounding

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Dielectric Optical Response

Classical Model

The dielectric function of a metal in a classical framework can be described by a simplified model where the conduction electrons are considered to constitute a gas with a number n of charges per volume unit, moving on a background of positive ion cores. Based on a classical motion equation for a carrier in a plasma under the influence of an external electric field and assuming a harmonic response, the dielectric function becomes [10, 11],

$$ \epsilon(\omega)=\epsilon_{\infty} - \frac{\omega_{p}^{2}}{\omega^{2}+i\gamma\omega}, $$
(3)

where 𝜖 ∞ corresponds to the interband screening contribution from the core electrons, while the term γ=1/τ is a damping frequency related to elastic collisions and depends on the relaxation time of the free electron gas τ. For its part, the plasmon frequency

$$ \omega_{p}^{2}=\frac{4\pi ne^{2}}{m^{*}}, $$
(4)

depends on the electronic density n, the elemental electric charge e, and the electron effective mass m ∗ [31].

These underlaying assumptions show how this so-called Drude model considers the relevant electrons as completely classical particles. However, despite such a limitation, that approach has been regularly used to describe plasmonic behavior in nanoparticles [32–36].

Quantum Model

Along with the size reduction of the metallic particles, the “continuous” conduction band of the nanostructure is expected to break up into well discretized states [9, 37].

As a first attempt to describe the optical response of the electron gas in a metal nanoparticle, Genzel et al. in ref. [37] presented a quantum model where electrons in the conduction band remained non-interacting, but experienced confinement by a hard-wall potential. Within this model, the dielectric function is given by

$$ \epsilon_{r} (\omega)=\epsilon_{\infty}+\frac{\omega_{p}^{2}}{N}\sum\limits_{i,f}\frac{s_{if}(F_{i}-F_{f})}{\omega_{if}^{2}- \omega^{2}-i\omega \gamma_{if}}, $$
(5)

where N is the total number of valence electrons in the nanoparticle and 𝜖 ∞ is the same as in the classical case. s i f , \(\omega _{if} \equiv \frac {E_{f} - E_{i}}{\hbar }\), and γ i f , are the oscillator strength, frequency, and damping for the dipole transition from an initial state ∣i〉 to a final state ∣f〉, respectively (F i and F f account for the corresponding values of the Fermi-Dirac distribution function).

In particular, the oscillator strength terms are the components that account for the effects of energy discretization, and include the geometrical features of the confinement. They are related to the dipole moment in the polarization direction (which in this study is chosen to be z) [38], according to

$$ s_{if}=\frac{2 m^{*} \omega_{if}}{\hbar}|\langle f|z|i \rangle |^{2}. $$
(6)

Following the geometry of the nanoparticles, we consider the infinite spherical potential well, as done in reference [8]. However, we go beyond the asymptotic approximation used in that work, by taking the full solutions of the Schrödinger equation for such a confining potential, i.e., wave functions of the form

$$ \psi_{n,l,m}(r,\theta,\phi)=\frac{1}{|j_{l+1}(\alpha_{nl})|} \sqrt{ \frac{2}{R^{3}}} j_{l} \left( \frac{\alpha_{nl}}{R}r \right){Y_{l}^{m}}(\theta,\phi), $$
(7)

where j l represents the lth spherical Bessel function, \({Y_{l}^{m}}\) the standard spherical harmonics, and α n l is the nth zero of j l (i.e. j l (α n l )=0 for n=0,1,2,…) [39].

Correspondingly, the discretized eigenenergies E n,l are related to the zeros of the spherical Bessel functions by the expression

$$ E_{n,l}=\frac{\hbar^{2}\alpha_{nl}^{2}}{2mR^{2}}. $$
(8)

By using z = rcos𝜃, and the wave functions from Eq. 7, the dipole moment in Eq. 6 can be obtained through the integral

$$\begin{array}{@{}rcl@{}} |\langle f|z|i\rangle|&=&{\int}_{0}^{2\pi}{\int}_{0}^{\pi}{{\int}_{0}^{R}}r^{2}\sin\theta dr d\theta d\phi\\ &&{\Psi}_{n_{f},l_{f},m_{f}}^{*}(r,\theta,\phi)r \cos \theta {\Psi}_{n_{i},l_{i},m_{i}}(r,\theta,\phi), \end{array} $$
(9)

whose angular and radial parts become, respectively

$$\begin{array}{@{}rcl@{}} I_{ang}&=&\sqrt{\frac{(l_{i}+m_{i}+1)(l_{i}-m_{i}+1)}{(2l_{i}+1)(2l_{i}+3)}}\delta_{\Delta l,+1}\\ &&+\sqrt{\frac{(l_{i}+m_{i})(l_{i}-m_{i})}{(2l_{i}+1)(2l_{i}-1)}}\delta_{\Delta l,-1},\\ I_{rad}&=&\frac{1}{|j_{l_{f}+1}(\alpha_{n_{f}l_{f}})|} \frac{1}{|j_{l_{i}+1}(\alpha_{n_{i}l_{i}})|}\left( \frac{2}{R^{3}}\right)\\ &&{{\int}_{0}^{R}}dr j_{l_{f}} \left( \frac{\alpha_{n_{f}l_{f}}}{R}r \right)r^{3} j_{l_{i}} \left( \frac{\alpha_{n_{i}l_{i}}}{R}r \right).\\ \end{array} $$
(10)

Thus, the integral reduces to

$$\begin{array}{@{}rcl@{}} |\langle f|z|i\rangle|&=&\frac{I_{ang}}{|j_{l_{f}+1}(\alpha_{n_{f}l_{f}})||j_{l_{i}+1}(\alpha_{n_{i}l_{i}})|}\left( \frac{2}{R^{3}}\right)\\ &&{{\int}_{0}^{R}}dr j_{l_{f}} \left( \frac{\alpha_{n_{f}l_{f}}}{R}r \right)r^{3} j_{l_{i}} \left( \frac{\alpha_{n_{i}l_{i}}}{R}r \right), \end{array} $$
(11)

which is different of zero only for values that satisfy Δl = l f −l i =±1.

Equations 8 and 11 allow for calculating the s i f terms, conducting to obtain via Eq. 5 a dielectric function that incorporates quantum features originated in size reduction.

Results

For silver isolated nanoparticles of radii ranging from 1 to 10 nm, we compute the real and imaginary parts of the dielectric function 𝜖 r (ω), and input it in Eq. 1 so that the FEF can be obtained.

In solving numerically the Helmholtz equation, the nanospheres were divided in a mesh of size decreasing tetrahedral elements, until convergence was achieved. The dielectric function within each considered model was calculated in frequency steps of 1013 Hz.

We carry out such a calculation in both frameworks, classical and quantum, by using Eqs. 3 and 5, respectively.

Within the classical approach, we take bulk optical constants from experiments by Johnson and Christie [40], whereas for the quantum approaches we use material parameters from Scholl et al. [8].

Figure 2a, b shows the real (imaginary) part of the dielectric function obtained from each of the considered approaches.

Fig. 2
figure 2

a Real part of the relative permitivity 𝜖 1(ω), obtained from both of the studied approaches. b Imaginary part of the relative permitivity 𝜖 2(ω), obtained from both of the studied approaches

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There are two remarkable differences between the dielectric functions obtained through these two approaches: First, the appearance in the quantum case of a fine structure due to inclusion of multiple transitions between discretized states. Second, also for the quantum case, the blue-shift of the main resonance with size-reduction [37], which can be understood in terms of the increased energy separation between eigenstates when the confinement is stronger [the fact that there is only one dielectric function in the classical framework is directly related to the size insensitivity of Eq. 3].

Figure 3a and b presents the obtained values for the field enhancement factor computed right on the north pole of the spherical nanoparticle, as calculated for the dielectric functions showed in Fig. 2a and b. There, it can be observed how the magnitude and activation energy of the FEF calculated by either the classical or the quantum approach, exhibit eye-catching differences. First, size dependence of the main plasmonic resonance in the quantum case appears in strong contrast to the stable peak position observed in the classical case. Second, an evident discrepancy in the intensity of the field enhancement is found.

Fig. 3
figure 3

a Field enhancement factor for silver nanospheres embedded in air, where 𝜖 r (ω) from the classical approach is used. b As in (a) but from the quantum approach

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Clearly, the insertion of quantum confinement effects in the dielectric function underlies these dissimilarities.

On one side, the size-dependent resonance is straightforwardly related to the enlargement of the energy separation between discretized states with increasing confinement, which is in complete agreement with results of absorption spectroscopy in colloids of silver nanoparticles [41], and EELS experiments on single metal nanoparticles [8]. On the other hand, the underestimation caused by using the classical approach should be associated to the fact of averaging the plasma contribution along all the spheric volume of the nanoparticle, while the quantum approach includes the wave function distribution, that concentrates close to the surface for excited states (the ones contributing the most to the summation in Eq. 5) [13].

Moderate underestimation of the FEF obtained within the Drude model is in agreement with time-dependent DFT calculations reported by Negre et al. in [42], but opposite to results from Zuloaga et al. in [10]. We would like to point out that the later work considers a damping constant around one order of magnitude larger than the one used in our calculations [8, 43], increasing so the imaginary part of the dielectric function and inflating the collective response in the nanoparticle.

Table 1 shows the relative underestimation of the enhancement factor in the classical framework (FEF C ) as compared to that in the quantum framework (FEF Q ).

Table 1 Relative underestimation for the different studied nanospheres
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It is noteworthy that the proportional difference in the magnitude of FEF for the compared approaches is neither maximum for the smallest nor the largest studied nanoparticles. This is important because it should be expected that at some radius larger than 10 nm, corrections associated with confinement effects start becoming negligible. Such non-monotonic behavior is here explicated in terms of a competition between the confinement effect (that is related to the surface to volume ratio and scales as \(\frac {1}{r}\)), and the number of available carriers constituting the stimulated plasmons (that is proportional to the number of atoms, accounts for the increase of FEF with the nanosphere size in both approaches, and scales as r 3).

As for the resonance frequency, one could at principle expect that for the largest simulated sphere the peak position would recover the classical limit, which is not observed in Fig. 3. It is important to mention that in the regime of “large” nanospheres (R = 10–100 nm), some other effects as dynamical surface screening and Lorentz friction become relevant [13, 30, 44], and their interplay alongside with softened quantum confinement contribute to a non-monotonic behavior for the plasmon resonance, before the classical limit is actually reached in micro-metric particles [8, 45].

Figure 4a and b presents the squared scattered fields normalized to the magnitude of the incident field at the vicinity of the nanosphere of radius R=1 nm, as obtained from the classical and quantum approaches, respectively. The shown 2D distributions are calculated in the corresponding resonance frequencies (as extracted from Fig. 3a and b).

Fig. 4
figure 4

Distribution of the near field enhancement factor in the nanospheres vicinity. a Obtained with the classical dielectric function, and b obtained with the quantum dielectric function

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These graphs are consistent with previous works in which the dipole plasmon mode is found to be by far the main contribution to the field enhancement in symmetric structures [46, 47]. Additionally, they help to visualize how the LSPR is strengthened within a model in which the wave functions of the carriers constituting the plasma are considered.

It is pertinent to mention that the general outline of our calculation is suitable for other geometries, as long as an appropriate confining potential is used. However, for arbitrary particle shapes, to deal with hundreds of numerical solutions of the eigenvalue problem for the summation in Eq. 5, might result a challenge.

Finally, we would like to address the fact that in our model, for the sake of simplification, an infinite confinement potential is considered implying that the widely discussed spill-out effect [13, 48, 49], is neglected. We would expect no qualitative change in our results if a finite potential was used, given that the physics underlying the resonance blue-shift and the classical underestimation of the FEF is not particularly sensitive to the potential height [37]. However, to establish how significant would be the quantitative change in case the potential allows for non vanishing electron density beyond the sphere edge, constitutes an interesting extension of this work.

Summary and Conclusions

We studied numerically the effects of quantum confinement on the field enhancement factor of metallic nanoparticles stimulated by time-dependent electromagnetic fields of varying frequencies. Remarkable differences in this optical response when calculated within a classical and a quantum framework were observed.

Influence of the surface to volume ratio and total number of conduction carriers on the collective excitation has been illustrated, and the need for including quantum phenomena in correctly describing the properties of small metallic nanoparticles is evidenced.

According to our results, significant underestimation in magnitude, and spurious insensibility of the plasmonic-induced resonance to the nanoparticle size, should be expected if a classical approach is used to study the field augment in spheres with radii at the order of few nanometers.

Furthermore, the fact that the discrepancies found in the field enhancement were originated in the dielectric function used for each approach suggest that alike effects could be anticipated in diverse geometries and in other optical responses (e.g., higher harmonic generation and frequency filtering).

We trust these findings will contribute to the optimization of ensembles for field detection and/or measurements, and somehow advance the understanding on the cross-over between classical and quantum regimes in nanostructures.