- Nano Express
- Open Access
Properties of Longitudinal Electromagnetic Oscillations in Metals and Their Excitation at Planar and Spherical Surfaces
- Vitaly V. Datsyuk^{1}Email author and
- Oleg R. Pavlyniuk^{1}
- Received: 30 December 2016
- Accepted: 12 July 2017
- Published: 1 August 2017
Abstract
The common definition of the spatially dispersive permittivity is revised. The response of the degenerate electron gas on an electric field satisfying the vector Helmholtz equation is found with a solution to the Boltzmann equation. The calculated longitudinal dielectric function coincides with that obtained by Klimontovich and Silin in 1952 and Lindhard in 1954. However, it depends on the square of the wavenumber, a parameter of the vector Helmholtz equation, but not the wave vector of a plane electromagnetic wave. This new concept simplifies simulation of the nonlocal effects, for example, with a generalized Lorents–Mie theory, since no Fourier transforms should be made. The Fresnel coefficients are generalized allowing for excitation of the longitudinal electromagnetic waves. To verify the theory, the extinction spectra for silver and gold nanometer-sized spheres are calculated. For these particles, the generalized Lorents–Mie theory gives the blue shift and broadening of the plasmon resonance which are in excellent agreement with experimental data. In addition, the nonlocal theory explains vanishing of the plasmon resonance observed for gold spheres with diameters less than or equal to 2 nm. The calculations using the Klimontovich-Silin-Lindhard and hydrodynamic dielectric functions for silver are found to give close results at photon energies from 3 to 4 eV. We show that the absolute values of the wavenumbers of the longitudinal waves in solids are much higher than those of the transverse waves.
Keywords
- Nonlocal electrodynamics
- Lindhard dielectric function
- Surface plasmon resonance
Background
Irradiation of a plane metal surface by femtosecond laser pulses often results in formation of laser-induced periodic surface structures (LIPSSs) [1]. Besides the LIPSS, hyperfine ripples called high-spatial-frequency LIPSS (HSFL) were observed [1, 2]. The spatial periods of the HSFL are significantly smaller than the irradiation wavelength λ _{0}. For example, for aluminum this period was estimated to range from 20 to 200 nm at λ _{0}=0.8 μm [2, 3]. While orientation of the ripples in ordinary LIPSS was perpendicular to the laser light polarization, the orientation of HSFL was often perpendicular and sometimes parallel to the polarization. Similar HSFL were formed on the surfaces of transparent dielectrics, semiconductors, and metals. The origin of the HSFL was explained by different mechanisms such as second-harmonic generation, the involvement of specific types of plasmon modes, self-organization, and local field enhancements during inhomogeneous breakdown in dielectric materials [2, 3].
The goal of this study is to search a wave process which could produce a pattern with a short period Λ≪λ _{0}. We examine properties of longitudinal (L) electromagnetic waves in metals also known as plasma waves. Our study consists of the following novel steps. First, we started our research with definition of the spatial dispersion of the permittivity. As shown below, the common definition is useless if a medium under study is not uniform and infinite. Therefore, we propose a new concept of the spatially dispersive dielectric function ε. This function establishes the direct proportionality between two vector fields, E(r,ω) and D(r,ω), but not the amplitudes E(k,ω) and D(k,ω) of plane waves. Consequently, the quantity ε depends on the square of the wave number, k ^{2}, the parameter of the vector Helmholtz equation for the electric field E(r,ω), but not the wave vector k of the plane waves. Then, to derive such a novel function, we determined the response of the conduction electrons on an electromagnetic mode by solving the Boltzmann transport equation written in the relaxation-time approximation. The so-called transverse and longitudinal Lindhard dielectric functions were obtained. Further, we found that the longitudinal Lindhard and much simpler hydrodynamic function are close in a wide range of parameters. Light extinction by silver and gold nanospheres was considered in order to illustrate the theory. We show for the first time that the nonlocal Mie theory explains the blue shift, broadening, and eventual vanishing of the plasmon resonances observed with a decrease of the size of the noble-metal nanospheres. Finally, the newly developed theoretical model was applied to examine the possibility of involvement of the longitudinal modes in formation of the laser-induced surface structures. For this purpose, we modified the Fresnel theory taking into account transmitted longitudinal waves.
Methods
where Δ is the Laplace operator.
As usual, the tangential components of the electric E and magnetic H fields are continuous across the boundaries of the media. In addition, we took into account that electrons are confined in metal; therefore, the following additional boundary condition (ABC) for the normal component of the current density j at the metal surface S was used: (j n)|_{ r∈S }=0.
where f is the single-particle distribution function in the phase space (r, v), v is the microscopic electron velocity, e and m are the electron charge and mass respectively, B is the magnetic induction, f _{0} is an equilibrium distribution function, and τ is the relaxation time.
Below, we derive formulas for the spatially dispersive dielectric functions. Then, we use them to study light reflection from a plane metal surface and scattering of light on a noble-metal nanosphere.
Results and Discussion
Spatial Dispersion of ε in a Heterogeneous Medium
where a spatially dispersive ε(ω, k) depends on the wave vector k of a plane electromagnetic wave. In our opinion, Eq. (3) is not ambiguous only in an infinite homogeneous volume but we deal with piecewise heterogeneous system where boundaries should be taken into account and k are not the same in different media.
Here E(ω, r) denotes distribution of the electric field but not merely the vector E in point r.
Longitudinal and Transverse Dielectric Functions
\(a=\frac {k v_{\mathrm {F}}}{\omega +i \Gamma }\).
In the case of Γ=0, this formula takes the form of an equation derived by Klimontovich and Silin [8] who studied Landau dumping in degenerate plasma (see [9], [10, Eq. (40.17)], and [11]). The permittivity of the equivalent Eqs. (16), (18), and (20) is commonly called the Lindhard dielectric function (with reference to [12]) though this function was first obtained by Klimontovich and Silin [8].
Reflection of a Plane Electromagnetic Wave from a Flat Metal Surface
Boundary Conditions
In this section, we determine the direction of the wave vector k ^{L} and amplitude of the L wave excited in metal during reflection of a plane electromagnetic wave from a flat metal surface.
were \(\hat {\mathbf {z}}\) is the unit vector in the direction of z axis.
where indexes 1x and 2x denote the x-projections of the vectors in media 1 and 2, respectively, θ _{1} is the angle of incidence.
Reflection and Transmission Coefficients
for the transmitted longitudinal wave, here r is a reflection coefficient, t and t _{L} are transmission coefficients.
At δ=0, the coefficient r becomes the Fresnel coefficient of reflection of the p-polarized wave (see, for instance, Eq. (2.49) of [4]). Under the same condition, t is not the Fresnel transmission coefficient since our definitions of t and r differ from the Fresnel’s ones.
Extinction of Light by Metal Nanosphere
ψ _{ l } and ζ _{ l } are the Riccati-Bessel and Riccati-Hankel functions of the order l, respectively; j _{ l } is the spherical Bessel function, the prime denotes the derivative of a function with respect to its argument.
The theoretical spectra in Fig. 2 were calculated using the Klimontovich-Silin-Lindhard and much simpler hydrodynamic dielectric functions. It is surprising that both calculations gave close results even though |a|>1 in the region of the plasmon resonance.
Resonant frequencies and widths of the dipolar plasmon resonance of single silver particles and particles’ beam
Size | ω _{m} (eV) | Δ ω (eV) | ||||||
---|---|---|---|---|---|---|---|---|
THEORY | ||||||||
Local | Nonlocal | Local | Nonlocal | |||||
A=0 | A=0.25 | A=0 | A=0.25 | A=0 | A=0.25 | A=0 | A=0.25 | |
D=4nm | 3.50 | 3.51 | 3.61 | 3.61 | 0.23 | 0.36 | 0.20 | 0.26 |
D=3 nm | 3.50 | 3.51 | 3.65 | 3.65 | 0.23 | 0.39 | 0.18 | 0.26 |
D=2 nm | 3.51 | 3.51 | 3.74 | 3.74 | 0.23 | 0.44 | 0.15 | 0.28 |
D=1 nm | 3.51 | 3.51 | – | – | 0.23 | 0.58 | – | – |
〈D〉=2 nm | 3.50 | 3.51 | 3.68 | 3.68 | 0.23 | 0.42 | 0.25 | 0.33 |
EXPERIMENT | ||||||||
〈D〉=2 nm | 3.65 | 0.33 |
In experiment [15], the peak frequencies ω _{m} and resonance widths Δ ω of the extinction spectra were almost independent of 〈D〉. This feature of Δ ω seems to disagree with the classical Mie theory. Really, the local theory predicts broadening of the plasmon resonances with the decrease in D (at A=0.25) as shown in Table 1. At the same time, the nonlocal theory gives approximately equal resonance widths but different peak positions. Superposition of the contributions from all particles gives the value of Δ ω which are in remarkable agreement with the experimental data. It is interesting that the nonlocal theory predicts a broadening of the plasmon resonance of a beam even at A=0.
At ω>4 eV, the smooth theoretical curves in Fig. 2 lie higher than the mash of narrow closely located experimental peaks. The interband absorption dominates in this spectral range as can be confirmed by Fig. 1. The observed peculiarities of the spectrum are likely to be a consequence of a transition from the continuum bands to a discrete level structure. Such a quantum-size effect was discovered earlier in a study of the optical properties of gold nanospheres [19]. When the silver-sphere size was increased to 〈D〉=3.5 nm, the absorption first increased relative to the maximum and formed a plateau with a series of small equidistant dips. Then, the absorption slightly decreased at 〈D〉=4 nm.
Wave Numbers of the Longitudinal Waves
The longitudinal modes differ from the transverse ones by much higher values of the wavenumbers. For example, for the calculations presented in Fig. 2, the real part of \(k_{2}^{\mathrm {L}}\) corresponds to the spatial period \(\Lambda =2 \pi /\Re k_{2}^{\mathrm {L}}\) decreasing from 9 to 2 nm at ω increasing from 3 to 4 eV. In this ω interval, the absolute value of the ratio \(k_{2}^{\mathrm {L}}/k_{2}\) decreased from 130 to 100 and the parameter δ of Eq. (27) decreased from 0.01 to 0.005 at θ _{1}=π/4. We conclude, therefore, that excitation of the L waves at a flat silver surface can be neglected. However, the L modes have been found to be of importance in nanometer-sized silver clusters.
In the simplest case of ε _{g}=1 and Γ=0, Eq. (35) predicts that metal is transparent for both transverse and L waves at ω>ω _{p} but both k ^{L} and k ^{T} are complex at ω<ω _{p}.
If solid is transparent, a longitudinal wave can be excited under oblique incidence of a p-polarized wave on a plane surface. There are several distinct features of this effect. First, the longitudinal waves can be generated at a flat surface, whereas special efforts should be made to excite the surface plasmon polaritons [4, 5]. Secondly, in the interference pattern, the electromagnetic-field intensity is modulated not along but perpendicular to the interface. Therefore, voids can appear in planes parallel to the surface due to spallation of the solid. According to the definition of ω _{p}, condition ω>ω _{p} can be met in solids (for example, semiconductors) with a low density of the current carriers. We do not examine this case here because the formula of ε ^{L} was derived for degenerate electron gas.
Conclusions
In order to define a spatially dependent dielectric function, all previous researchers considered interaction of matter with a plane electromagnetic wave. This approach is not constructive and rigorous in nano-optics when the field is localized in a cavity and the boundary conditions must be somehow taken into account. We have solved this problem by calculating the response of the medium on an electric field that satisfies the vector Helmholtz equation. The derived spatially dispersive dielectric function depends on the square of the wavenumber, a parameter of the Helmholtz equation, but not the wave vector of a plane wave.
We report the Fresnel reflection coefficients modified due to excitation of the longitudinal waves in metals. Similar generalization was made earlier for the Mie coefficients. Herein, the theory has been verified with simulation of light extinction by nanometer-sized silver and gold clusters. The calculated shift from 3.5 to 3.65 eV and the width of the surface plasmon resonance of the silver particles’ beam are in excellent agreement with the experimental data. In addition, the nonlocal model explains the vanishing of the plasmon resonance of golden spheres with diameters of about 2 nm. It is important that L wave can be excited on a flat surface by a plane incident wave. This is the main difference of the plasmon polaritons from the surface plasmon polaritons.
The properties of the electromagnetic oscillations in metals have been examined. It has been found that the absolute values of the wavenumbers of the longitudinal waves are much larger than those of the transverse waves. For example, in silver at a photon energy of 3.5 eV, the ratio of the absolute values of the wavenumbers is equal to 130. There, the real part of the wavenumber of the longitudinal wave corresponds to a wavelength of 7 nm. The large difference in the wavenumbers prevents excitation of the L waves at a planar surface. However, the L modes have been shown to be excited in silver and gold nanometer-sized particles.
Declarations
Competing Interests
The authors declare that they have no competing interests.
Authors’ contributions
In this study, the researchers played slightly different roles. VVD derived the longitudinal dielectric function. ORP focused on excitation of the longitudinal modes in various systems with particular attention paid to the LIPSS. Both authors read and approved the final manuscript.
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References
- Bonse J, Höhm S, Kirner SV, Rosenfeld A, Krüger J (2017) Laser-induced periodic surface structures–A scientific evergreen. IEEE J Sel Top Quantum Electron23(3): 109–123.View ArticleGoogle Scholar
- Bonse J, Krüger J, Höhm S, Rosenfeld A (2012) Femtosecond laser-induced periodic surface structure. J Laser Appl24(4): 042006.View ArticleGoogle Scholar
- Bashir S, Rafique MS, Husinsky W (2012) Femtosecond laser-induced subwavelength ripples on Al, Si, CaF _{2} and CR-39. Nucl Instr Meth Phys Res B275: 1–6.View ArticleGoogle Scholar
- Novotny L, Hecht B (2006) Principles of, Nano-optics. Cambridge University Press, New York.View ArticleGoogle Scholar
- Maier SA (2007) Plasmonics: Fundamentals and Applications. Springer, New York.Google Scholar
- García de Abajo FJ (2008) Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides. J Phys Chem C112(46): 17983–17987.View ArticleGoogle Scholar
- Kliewer KL, Fuchs R (1969) Lindhard dielectric functions with a finite electron lifetime. Phys Rev181(2): 552–558.View ArticleGoogle Scholar
- Klimontovich YL, Silin VP (1952) On the spectra of systems of interacting particles, (in Russian). Zh Ehksper Teor Fiz23: 151–160.Google Scholar
- Klimontovich YL, Silin VP (1960) The spectra of systems of interacting particles and collective energy losses during passage of charged particles through matter. Phys.-Uspekhi3(1): 84–114.View ArticleGoogle Scholar
- Liftshitz EM, Pitaevskii LP (1997) Physical Kinetics. Butterworth–Heinemann, Oxford.Google Scholar
- Larkin IA, Stockman MI, Achermann M, Klimov VI (2004) Dipolar emitters at nanoscale proximity of metal surfaces: Giant enhancement of relaxation in microscopic theory. Phys Rev B69(12): 121403–11214034.View ArticleGoogle Scholar
- Lindhard J (1954) On the properties of a gas of charged particles. Kgl Danske Vidensk Selsk Mat-Fys Medd28(8): 1–57.Google Scholar
- Ordal MA, Bell RJ, Alexander RW Jr, Long LL, Querry MR (1985) Optical properties of fourteen metals in the infrared and far infrared: Al, Co, Cu, Au, Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V, and W. Appl Opt24(24): 4493–4499.View ArticleGoogle Scholar
- Datsyuk VV (2011) A generalization of the Mie theory for a sphere with spatially dispersive permittivity. Ukr J Phys56(2): 122–129.Google Scholar
- Hilger A, Tenfelde M, Kreibig U (2001) Silver nanoparticles deposited on dielectric surfaces. Appl Phys B73(4): 361–372.View ArticleGoogle Scholar
- Lynch DW, Hunter WR (1985). In: Palik ED (ed)Handbook of Optical Constants of Solids, 355–356.. Academic Press, San Diego.Google Scholar
- Bohren CF, Huffman DR (1983) Absorption and Scattering of Light by Small Particles. Wiley, New York.Google Scholar
- Kreibig U, Vollmer M (1995) Optical Properties of Metal Clusters. Springer, Berlin.View ArticleGoogle Scholar
- Alvarez MM, Khoury JT, Schaaff TG, Shafigullin MN, Vezmar I, Whetten RL (1997) Optical absorption spectra of nanocrystal gold molecules. J Phys Chem B101(19): 3706–3712.View ArticleGoogle Scholar
- Johnson PB, Christy RW (1972) Optical constants of the noble metals. Phys Rev B6(12): 4370–4379.View ArticleGoogle Scholar
- Weber MJ (2002) Handbook of Optical Materials. CRC Press, Boca Raton.View ArticleGoogle Scholar
- Hao F, Nordlander P (2007) Efficient dielectric function for FDTD simulation of the optical properties of silver and gold nanoparticles. Chem Phys Lett446(1): 115–118.View ArticleGoogle Scholar
- Drachev VP, Chettiar UK, Kildishev AV, Yuan HK, Cai W, Shalaev VM (2008) The Ag dielectric function in plasmonic metamaterials. Opt Express16(2): 1186–1195.View ArticleGoogle Scholar