Continual–quantum plasmonics with kinematical functions: dipolar resonance and nonlocal polarizability of simple metal made nanoparticles

Abstract

In this work we formulate the theory of a mesoscopic oscillator equivalent to sp-electron excitations in simple metal made nanoparticles illuminated by incident light. The principles of continuum mechanics have been applied to maintain this goal: the primary hypotheses about the dependences of electron density function upon kinematical generalized coordinates, the stationary action principle, and the perturbation method. On their grounds, dynamic equations describing the motion of electron gas subject to alternating potentials together with the ground state equations have been derived. A methodological advantage of the latter is in the correct (qualitative and quantitative) prediction of Friedel oscillations and electron spill-out through an ion lattice with no demands to use high power computer resources as opposed to the orbital density functional theory. The dynamic equations allow studying of the nanoparticle resonant properties in an analytical form without need of numerically solving them. It has been shown with their use that the resonant frequency of the main dipolar resonance becomes the density functional of the ground state. On the basis of the dynamic equations, the theory of nonlocal polarizability has been deduced that does not impose the homogeneity (or weak inhomogeneity) constraints on the electron gas density. In this context the two following results are of importance. We manage to: (1) to demonstrate the effect of giant nonlocality of the dipole moment—its formation by the events separated by distances significantly larger than nanoparticle dimensions and temporal intervals much larger than the mean free time; (2) to derive the expression of the volumetric mode of compression-tension that is resonant on frequency of the main dipolar resonance.

Appendices

Appendix 1

Let the vector and scalar potentials be defined accurate within the gradient of an arbitrary function and its temporal derivative as: ${\mathbf{A^{\prime}}} = {\mathbf{A}} + \nabla f$, ${\psi^{\prime}} =\uppsi - {{\partial f} \mathord{\left/ {\vphantom {{\partial f} {\partial t}}} \right. \kern-0pt} {\partial t}}$, respectively. Substituting of ${\mathbf{A^{\prime}}}$ and ${\psi^{\prime}}$ into the interaction energy ${\mathbf{j}}_{4} \cdot {\mathbf{A}}_{4}$ leads to the appearance of the term:

$$\begin{aligned} & \mathop {\lim }\limits_{{diam\;{\varvec{\Omega}} \to \infty }} \int_{{t_{1} }}^{{t_{2} }} {\int_{{\varvec{\Omega}}} {\left\{ {{{\partial (\uprho_{tot} \,f)} \mathord{\left/ {\vphantom {{\partial (\uprho_{tot} \,f)} {\partial t}}} \right. \kern-0pt} {\partial t}} + \nabla \cdot ({\mathbf{j}}_{tot} \,f)} \right\}\;d{\mathbf{r}}\,dt} } \\ & \quad = \int_{{\,R^{3} }} {\left. {(\uprho_{tot} \,f){\kern 1pt} } \right|_{{t_{1} }}^{{t_{2} }} \;d{\mathbf{r}}} + \mathop {\lim }\limits_{diam\;\varSigma \to \infty } \;\int_{{t_{1} }}^{{t_{2} }} {\oint_{\,\varSigma } {{\mathbf{n}} \cdot\uprho_{tot} {\mathbf{v}}\,f\;d{\mathbf{r}}\,dt} } \\ \end{aligned}$$

(23)

inside the action S, where ${\mathbf{j}}_{tot} = {\mathbf{j}}_{e} + {\mathbf{j}}_{e}^{src}$ and $\uprho_{tot} =\uprho_{e} +\uprho_{e}^{ + } +\uprho_{e}^{src}$. The second term in the right hand side of (23) vanishes, because ${\mathbf{v}}$ is bounded and $\mathop {\lim }\nolimits_{{|{\mathbf{r}}| \to \infty }}\uprho_{tot} = 0$. The first term variation yields vanishing quantity, because $\uprho_{tot}$ is varied only at the extremities of the time interval $(t_{1} ,\;t_{2} )$ at which: $\delta {\mathbf{u}}({\mathbf{r}},t_{1} ) = \delta {\mathbf{u}}({\mathbf{r}},t_{2} ) = 0$. Therefore, the gauge dependence of ${\mathbf{A^{\prime}}}$ and ${\psi^{\prime}}$ does not add new terms to the action variation: $\delta S^{\prime} = \delta S$. Consequently, the motion equation remains unchanged. No new terms appear in the free EM field Lagrangian due to the gauge invariance of the fields ${\mathbf{E}}$ and ${\mathbf{B}}$. The variation of (23) over generalized coordinates ${\mathbf{A}}$ and $\uppsi$ yields null identically, because (23) is independent of them.

Appendix 2

In Eq. (7) the following energy functional has been used:

$$F = - \underbrace {{(3\uppi^{2} )^{{{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}} \frac{{3\hbar^{2} }}{10m}\;\uprho_{0}^{{{\raise0.7ex\hbox{$5$} \!\mathord{\left/ {\vphantom {5 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}} }}_{{{\rm T}homas - Fermi\;KE}} - \underbrace {{\uplambda_{vW} \frac{{\hbar^{2} }}{8m}\frac{{|\nabla\uprho_{0} |^{2} }}{{\uprho_{0} }}}}_{{von\;Weizs\ddot{a}cker\;KE}} + \underbrace {{\frac{{3e^{2} }}{{16\uppi \upvarepsilon _{0} }}\left( {\frac{3}{\uppi}} \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}}\uprho_{0}^{{{\raise0.7ex\hbox{$4$} \!\mathord{\left/ {\vphantom {4 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}} }}_{Dirac\;(LDA)\;exchange\;energy \, } + \underbrace {{0.0333\,E_{h} \;\uprho_{0} \,g_{GL} (x)}}_{\begin{subarray}{l} Correlation\;correction\;of \\ Gunnarsson\;and\;Lundqvist \end{subarray} },$$

where $\uplambda_{vW}$ is the von Weizsäcker parameter, $E_{h}$-the Hartree energy, $x = \frac{1}{{11.4\,a_{B} }}\left( {\frac{3}{{4\uppi \uprho _{0} }}} \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}}$-a dimensionless complex; $g_{GL} (x) = \left( {1 + x^{3} } \right)\ln \left( {1 + x^{ - 1} } \right) - x^{2} + ({1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2})x - {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}$ (Ekardt 1984; Gunnarsson and Lundqvist 1976; Yannouleas et al. 1993; Banerjee and Harbola 2000). The coefficient $\uplambda_{vW}$ in the von Weizsäcker definition is unity (Parr and Yang 1989). The density matrix expansion on powers of $\hbar$ yields the value: $\uplambda_{vW} = {1 \mathord{\left/ {\vphantom {1 9}} \right. \kern-0pt} 9}$ (Parr and Yang 1989). In the Thomas–Fermi–Dirac–von Weizsäcker theory of energy functionals $\uplambda_{vW}$ is considered to be an empirical quantity localized inside the interval $({1 \mathord{\left/ {\vphantom {1 9}} \right. \kern-0pt} 9},1)$; its exact value is unknown and there exist different recommendations on how to choose it (Yan 2015; Li et al. 2015; Parr and Yang 1989; Chan et al. 2001).