Electronic spill-out induced spectral broadening in quantum hydrodynamic nanoplasmonics

Xiaoming Li; Hui Fang; Xiaoyu Weng; Lichao Zhang; Xiujie Dou; Aiping Yang; Xiaocong Yuan

doi:10.1364/OE.23.029738

1. Introduction

Plasmon resonances can be regarded as collective oscillations of free charge carriers introduced by Coulomb interactions in conducting media [1]. They have the ability to confine the electromagnetic field to regions well below the diffraction limit with extremely high intensities [2], based on which a wide range of novel applications emerged in these decades, including various of nanophotonic devices [3,4 ], biomedical sensing [5], surface enhanced Raman spectroscopy [6], and optical trapping for metallic particles [7]. With the rapid development in nanoscale fabrication, these nanoplasmonic structures can be made with size down to only a few nanometer where the quantum mechanism effects such as nonlocal response and electronic spill-out [8–16 ] become dominated over the classical electrodynamics. In order to explain these phenomena, two main approaches are used: the density-function theory (DFT) [12,17–20 ] and the hydrodynamic theory (HT) [21–24 ]. In comparison, the semi-classical HT can be applied for relatively large plasmonic structures that are beyond practical computing power for the pure quantumical DFT.

The development of the hydrodynamic model has undergone two major phases. The first one is the emergence of the hydrodynamic Drude model. The key idea is to incorporate the Thomas-Fermi (TF) pressure into the equation of motion of the electron gas. This model can give accurate predictions of the spectral positions of the surface plasmon resnonances for noble-metal nanoparticles at size regimes where the pure classical electrodynamics is invalid [4, 10 ], and has also been employed to successfully explain the ultimate limits of field enhancement [21], and as well the fundamental limitations of Purcell factors in plasmonic waveguides [25]. One shortcoming of this model is that the electron spill-out effects is neglected, i.e., the density distribution of electron cloud outside the metal is set to be zero [22,23,26 ]. It is worth mention that recently there are several novel models have been proposed to circumvent this problem in the classical electrodynamics framework [27–32 ]. For instance, the quantum-corrected model can appropriately describe the quantum tunneling in the narrow gap between the metallic dimers [27,32 ].

Another significant leap for the improvement of the TH is the introduction of the von Weizsäcker (VW) term, which is the first-order correction to the TF kinetic energy term by considering the inhomogeneous lying in the carrier distribution [22,23 ]. In such extended HT (also called quantum-hydrodynamic theory, i.e. QHT), the striction of the hard-wall potential boundary condition is removed, and the electronic spill-out emerges naturally. At the same time, the exchange and correlation Coulomb corrections can also be added to further improve accuracy. QHT can accurately reflect the free carrier distribution both inside and outside of the metal. Recently, Giuseppe Toscano has applied this theory, obtaining the redshift optical response of Na nanowires and the blueshift optical response of Ag nanowires, which quantitatively agree with the experiments and the more advanced quantum theoretical calculation [22]. In addition, QHT keeps the high computational efficiency as that of the hydrodynamic Drude model. Nonetheless, there are still many open questions remaining for the current version of QHT. For example, the actual spectral width in different nano-plasma system do not show out, which is certainly another important parameter besides the spectral position.

In this paper, we take a step forward in improving QHT by taking the spectral broadening into account. On one hand, the Coulomb interaction is responsible for the plasma oscillations, an organized motion of a large number of electrons [1]. For a single electron, its spectral width is determined by the Coulomb scattering rate [33]. While for a whole nano-plasma system, the absorption spectrum width depends on the contributions from all of the free carriers. On the other hand, due to the spill-out, the electron density distribution is inhomogeneous, especially in the vicinity of the metal surface. Furthermore, different carrier densities result in different screening of the Coulomb interactions [34]. Thus, the effective Coulomb scattering rate of the single electron is not a constant for different carrier densities, leading to a varying spectral width according to the spatial position. Through the introduction of the relationship between the single-electron spectral width and the electron density, we study how the size of the nanostructures affects the broadening of the absorption spectrum, and then compare the numerical results with the reported experiment. These results demonstrate that our proposed method is very effective for predicting the spectral broadening in nano-plasma system, thus extends the current HT theory.

2. Theoretical model

In the framework of the QHT, the starting point is the semiclassical Hamiltonian for describing the inhomogeneous electron gas which can be written as [24]:

\begin{array}{l} H [n (r, t), p (r, t)] = G [n (r, t)] + \int \frac{{(p (r, t) - e A (r, t))}^{2}}{2 m} n (r, t) d r \\ \begin{matrix}  \end{matrix} + e \int ϕ (r, t) n (r, t) d r + e \int [\frac{e}{2} \int d r^{'} \frac{n (r^{'}, t)}{4 π ε_{0} | r - r^{'} |} - V_{i o n} (r)] n (r, t) d r, \end{array}

where the dependent variables n(r,t) is the density of the electron gas, and p(r,t) = mv(r,t) + eA(r,t) is the canonical momentum of an electron in electromagnetic fields with e the electron charge and m the electron mass. Here, A(r,t) and ϕ(r,t) are the vector potential and scalar potential induced by the electromagnetic fields, respectively. The term G[n(r,t)] is the internal quantum energy of the electron gas, V _ion(r) denotes the electrostatic potential from the positive ion background with a constant density n _ion inside the metal. ε₀ is the permittivity of free space. In general, the internal energy functional G can be expressed as [23,24 ]

\begin{array}{l} G = \frac{3}{10} \frac{ℏ^{2}}{m} (^{3 π^{2}) 2 / 3} \int n^{5 / 3} (r, t) d r + \frac{λ_{ω}}{8} \frac{ℏ^{2}}{m} \int \frac{{| \nabla n (r, t) |}^{2}}{n (r, t)} d r \\ - 0.0588 \frac{e^{2}}{ε_{0}} \int n^{4 / 3} (r, t) d r - \frac{e^{2}}{ε_{0}} \int \frac{0.035}{0.6024 + 7.8 a_{H} n^{1 / 3}} n^{4 / 3} (r, t) d r, \end{array}

which includes the internal kinetic energy terms (the first two terms) and the exchange-correlation energy terms (the last two terms). Here λ_ω represents the coefficient of the second-order inhomogeneity correction, and a _H the Bohr radius. The first term contributing to the internal kinetic energy is the Thomas-Fermi (TF) kinetic functional and the second term is the VW kinetic functional. We note that the VW kinetic energy term is the key part describing the electronic spill-out: it gives the leading-order correction to the kinetic energy induced by the rapidly varying densities around the metal surface.

The first step to characterize the linear-response dynamics (LRD) of an inhomogeneous electron gas in metallic media is to obtain the equilibrium electron density n ₀, which is a solution of the following equation [22]:

\nabla^{2} {(\frac{δ G}{δ n})}_{0} - \frac{e^{2}}{ε_{0}} n_{0} + \frac{e^{2}}{ε_{0}} n_{i o n} = 0.

To proceed, we assume a weak external perturbation. Then the corresponding variables can be written as n(r,t) = n ₀(r,t) + n ₁(r,t), δG/δn = (δG/δn)₀ + (δG/δn)₁. The LRD due to the driving of the external electromagnetic field is determined by the coupling of the Maxwell’s wave equation and the single-electron motion equation, which are as follows written in the frequency domain [22]:

\nabla \times \nabla \times E_{1} - \frac{ω^{2}}{c^{2}} E_{1} = μ_{0} i ω J_{1}

\nabla \cdot J_{1} = \frac{1}{γ - i ω} [- \nabla \cdot \frac{e n_{0}}{m} \nabla {(\frac{δ G}{δ n})}_{1} + ε_{0} E_{1} \cdot \nabla ω_{p}^{2} + \frac{ω_{p}^{2}}{i ω} \frac{\nabla \cdot J_{1}}{ε_{r} (ω)}],

where ω _p = [e ² n _ion/(mε ₀)]^1/2 is the common plasma frequency of the electron gas, J ₁ is the polarization current in the frequency domain, and E₁ is the intensity of the polarization electric field. The other parameters include c the speed of light in vacuum, μ ₀ the permeability of vacuum, ε _r(ω) the screening permittivity due to the interband effects, and γ the damping rate of a single electron.

Here, the damping rate γ, which has been phenomenologically put into Eq. (5), is a key parameter that determines the width of the absorption spectrum for a nano-plasma system. We will show that the spectral widths for the whole system and the single electron are the same if γ takes a fixed value across the system, being independent of the other parameters. In other words, it is not possible to gain any information about the spectral width except the preset damping rate. As been addressed in the introduction, a better approach is to consider the damping rate as a function of the electron density. Here, we will apply the experience nonlinear relationship

γ [n (r), ε_{d} (r)] \propto n {(r)}^{- 5 / 6} ε_{d} {(r)}^{- 1 / 2},

which has been used in [34] in the context of discussing the experiment results of size-dependent electron thermalization time for Ag nanoparticles, where ε _d(r) represents the dielectric permittivity related to the core electron screening.

3. Results and analyses

By following the description found in [22, 23 ], we establish a COMSOL computation procedure to implement the above QHT model. Before first carrying out the calculation for the equilibrium electron density, the parameter λ_ω, the critical factor to determine the electronic spill-out, need to be predetermined. However, there currently has no well defined expression available for setting it, and in practical numerical simulation, a constant in the range of 1/9 to 1 [35] has to be chosen. In our calculation, we proposed a mixed setting of λ_ω instead of a single value, i.e.

λ_{ω} = λ_{m i x} = {\begin{cases} λ_{i n} \equiv 1 / 9, i nside of the metal, \\ λ_{o u t} \equiv 1 / 2, o u t s i d e o f t h e m e t a l . \end{cases}

We think this setting should better meet the real situation since the electron density varies slowly inside the metal nanostructure but very rapidly near the surface. We investigated the numerical outcome under this hypothesis for a Na cylindrical nanowire as shown in Fig. 1 , where the nanowire has a radius of R = 2nm and the electron spill-out-layer is set to have a thickness of R _im = 0.5nm. The ion density appeared in Eq. (3) takes the value of n _ion = 2.5173 × 10²⁸m⁻³ [22].

Fig. 1 Schematic diagram of the metal nanowire (left) and its sectional view (right). The radius of the nanowire (metal layer) is R, and the electron spill-out-layer is R _im.

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From numerically solving Eq. (3), the equilibrium electron densities as a function of spatial position with different values of λ_ω are plotted in Fig. 2 . To facilitate the comparison, we use n ₀/n _ion instead of n ₀ to express the distribution. The common feature of these results is in three folds: 1) the electron density in the metal layer is relatively stable which approaches the ion density n _ion; 2) it diminishes rapidly in the spill-out layer; 3) it undergoes a dramatic change near the metal surface. The differences corresponding to the different λ_ω settings are also apparent. For λ_ω = 1/9 as shown in Fig. 2(a), n ₀/n _ion is only about 2.7e-11 on the outer boundary of the spill-out-layer. When λ_ω is set to 1/4 in Fig. 2(b), the minimum value of n ₀/n _ion is about 1.3e-7. Figure 2(c) shows that this value increased to 1.5e-5 with λ_ω = 1/2. These results clearly demonstrate that the value of λ_ω controls the degree of the electron spill-out effect with the low value corresponding to a less spill-out. As for our setting with λ_mix, as shown in Fig. 2(d), because of the hybrid nature of the setting, the free electron density inside the nanowire is as stable as Fig. 2(a) while the density outside reflect the clear spill-out comparable to Fig. 2(c). As can be seen, the electron density variation near the surface becomes more pronounced due to the discontinuity of the setting values at the interface. The major purpose of the above analysis is to demonstrate that the choice of setting λ_ω will impact the electron spill-out in a dramatic fashion thus is a critical issue.

Fig. 2 The equilibrium electron density for n ₀/n _ion when R = 2nm. (a) λ_ω = 1/9; (b) λ_ω = 1/4; (c) λ_ω = 1/2; (d) λ_ω = λ_mix with λ_in = 1/9 and λ_out = 1/2.

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Next we focus on another important parameter γ which currently also lacks well-defined guidance for the setting. First, we simulated the absorption cross section spectra under the setting of γ = 0.17 eV according to [22]. The results for the different values of λ_ω are presented in Fig. 3 from which we can draw two major conclusions. One is that the smaller λ_ω is, the larger the blueshift of the resonance peak will be and the lower of the magnitude of the second resonance peak. For our setting of λ_mix, the spectrum shows only a blueshift about 0.07 eV relative to that for λ_ω = 1/9. For λ_ω = 1/9, the resonance frequency calculated by QTH is 4.17 eV, which agrees well with the result of 4.09 eV obtained from the time-dependent density-function theory [11]. In addition, there is a tailing peak around 3 eV for λ_ω = λ_mix. This phenomenon should be mainly caused by the discontinuity of the setting value of λ_ω near the metal surface. The other is that for all of the settings, as can be seen, all of the spectra widths of the major resonance peaks are almost the same, implying that setting γ as a constant might not be a good strategy if the accurate spectral width need to be reflected.

Fig. 3 Absorption spectra for a Na nanowire with radius R = 2 nm and γ = 0.17 eV when λ_ω = 1/9, λ_ω = 1/4, λ_ω = 1/2 and λ_ω = λ_mix, respectively.

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Finally, we take the effect of the Coulomb interaction screening, i.e., the function expressed in Eq. (6) into account. For simplicity in this paper, we neglect the screening from the d electron, i.e. to set ε _d = 1 both inside and outside of the metal. Assuming that γ[n _ion] = 0.17 eV, then γ _n = γ[n(r)] = 0.17 × (n ₀(r) /n _ion) ^-5/6 eV. The size-dependent spectral width for a Na nanowire is investigated.

From Fig. 4 we can see that in stark contrast to the results by setting γ = γ _c = 0.17 eV which showing a constant spectral width (0.17 eV) for all of the radiuses (the blue dotted line), the results by using γ _n = γ[n(r)] = 0.17 × (n ₀(r) /n _ion)^-5/6 eV (the red solid line) show a clear trend of nonlinear decreasing of the spectral width versus the nanowire radius. We found that this trend is consistent with the experiment results reported in [34]. This physical phenomenon can be accounted for by the following facts. First, the electron density is inhomogeneous everywhere with high degree near the metal surface due to electron density spill-out from a surface. Second, the Coulomb interaction screening is less efficient for smaller electron density at the spill-out region, leading to a higher electron-electron scattering rate. As a consequence, the spectral width of a single electron near the surface should be larger than that in the metal. Because when the radius of the nanowire becomes smaller, the surface effect will be enhanced, i.e., the contribution from the surface state electrons will occupy a high proportion, it is anticipated that the absorption spectral width and the nanowire radius will form a nonlinear relationship.

Fig. 4 Size dependence of the spectral width for a Na nanowire with λ_ω = λ_mix. Solid line indicates γ = γ _n; and dotted line for γ = γ _c.

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Based on the above studies, we propose a new form to describe the spectral width of a whole nano-plasma system (γ _p):

γ_{p} = \int f [n (r)] \cdot γ [n (r)] d r,

where f[n(r,t)] is a proportionality factor correlated to the geometric structure and material composition of the nano-plasma system. The major character of f[n(r,t)] is its nonlinearity which is caused by the electronic spill-out, leading to spectral broadening for different sizes. This formula also explicitly expressed that the plasma resonance is a collective behavior, rather than from a single electron. We note that the accuracy of the absorption spectra actually depends on the development both of the basic theoretical and experimental physics [36]. Only when the exact contribution of the VW kinetic functional, the exchange-correlation energy and the damping rate are known, the precise properties of the absorption spectra can be obtained.

4. Conclusion

In conclusion, we have first studied the impact of VW kinetic functional for getting the correct electron density distribution through propose a mixed function of λ_ω to meet practical situation. It turns out that the value of λ_ω is a critical factor to appropriately describe the effect of electronic spill-out. We then show that when γ is a constant for all electrons, different values of λ_ω and size can only change the resonance peak, while it can’t predict the spectral broadening. For this reason, we take the Coulomb interaction screening into consideration. And the results show that the width of the absorption spectrum becomes larger as the size decreases, which is consistent with the reported experiment. Finally, we get a conclusion that the absorption spectra for a nano-plasma system are determined by contributions from all electrons, which can be described clearly through a new form of plasma spectral width we proposed. Our theoretical studies thus extend the current quantum-hydrodynamic theory and we believe it will become a useful method to study the ultrafast dynamics in nano-plasma system.

Acknowledgments

This work is supported by the National Basic Research Program of China (Grant No. 2015CB352004) and the National Natural Science Foundation of China (NSFC) under Grant No. 61427819. X.Y. acknowledges support from the Science and Technology Innovation Commission of Shenzhen under grant No. KQCS2015032416183980, JCYJ20140418091413543, and the Start-up Funding at Shenzhen University.

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Electronic spill-out induced spectral broadening in quantum hydrodynamic nanoplasmonics

Abstract

1. Introduction

2. Theoretical model

3. Results and analyses

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (8)

Optics Express