Theory of Localized Plasmons for Multiple Metal Nanostructures in the Random Phase Approximation

A theory is derived for localized plasmons in multiple metal nanostructures by developing the theory of localized bulk and surface plasmons for metal nanostructures in the random phase approximation (RPA) at the high frequency condition. The local electron density in multiple metal nanostructures is expressed as the sum of the electron density of each metal nanostructure. Self-consistent integral equations derived in the RPA give determinants to calculate the localized surface plasmon frequencies for multiple metal nanospheres with step-function-like electron density at their surfaces. The frequencies are analytically calculated in the dipole approximation for a dimer and chain of metal nanospheres. The frequencies are redor blue-shifted depending on the spacing between the metal nanospheres. A light emission formula is also derived for multiple metal nanostructures in the dipole approximation. The light emission intensities from the dimer and chain are analytically calculated using the step-function-like electron density model. The retardation effect on the localized plasmons for multiple metal nanostructures is then investigated by applying the structural Green’s function method, which is used to calculate the electronic structures of condensed matter. [DOI: 10.1380/ejssnt.2015.391]


I. INTRODUCTION
Surface plasmons can couple with photons to produce collective excitations called surface plasmon polaritons (SPPs) and localized surface plasmon (LSPs) [1][2][3][4][5].SPPs and LSPs can concentrate optical waves into regions that are much smaller than their free space radiation wavelengths.They can also greatly enhance their local electric fields near metal nanostructures at the surface plasmon excitation energies.These effects make it possible to use SPPs and LSPs to fabricate nanoscale photonic devices and to perform optical measurements with extremely high detection sensitivity.It has recently been reported that the localized plasmons in multiple metal nanostructures reveal interesting phenomena, such as a red-shift of the surface plasmon frequency, large enhancement of the local electric fields, and the quantum tunneling effect between the metal nanostructures [6][7][8][9][10][11].
Many theoretical studies of SPPs and LSPs have been reported, including studies based on the hydrodynamic method [12][13][14][15], semi-classical method [16][17][18], random phase approximation (RPA) [19][20][21], and time-dependent local density approximation [22][23][24][25].Theoretical studies of the interaction between the external electric field or the incident electrons and the surface plasmon have been reported [26][27][28][29][30].However, these studies have mainly focused on surface plasmon excitation, and the coupling phenomena between bulk and surface plasmons, which play important roles in the plasmon excitation process, have not been extensively studied.The coupling is caused by the many-body interaction between the electrons in metal nanostructures.
The author has also developed a theory of localized plasmons using the RPA under high frequency conditions, where the coupling between the bulk and surface plasmons is properly considered in metal nanostructures [31].In this theory, the local electron density in the metal nanostructures plays an essential role in the plasmon excitation.This treatment has been developed to study the surface plasmon excitations and nonlinear optical responses in metal nanoclusters [32,33].The theory has been recently developed to calculate the light emission intensity from a single metal nanostructure by considering the retardation of the scalar potentials [34].
In this study, a theory of localized plasmons for multiple metal nanostructures is derived using the RPA at the high frequency condition by developing previous theories.Selfconsistent integral equations are derived to calculate the localized plasmon frequencies for multiple metal nanostructures.Some model calculations for metal nanospheres show red-or blue-shifts of the LSP frequencies depending on the spacing between the metal nanospheres.The light emission phenomena from the LSPs are also investigated for multiple metal nanostructures.The retardation effects on the LSPs in multiple metal nanostructures are then investigated by applying the structural Green's function method, which is used to calculate the electronic structures of condensed matter.
Volume 13 (2015) Ichikawa dition, i.e., the plasmon excitation condition: where ψ n (r 1 ) is the normalized single-electron wave function in the metal nanostructure, θ is the step function, and E F and E n are the Fermi energy and the single electron energy, respectively.n (r 1 ) is the local electron density in the metal nanostructure, m e , −e are the electron mass and charge, and ∇ 1 represents the gradient with respect to r 1 .G 0 (r − r 1 , ω) is the retarded Green's function, which satisfies the following Helmholtz equation in Gaussian units: where ) and c is the velocity of light.The derivation of Eq. ( 1) is described in Appendix A. Integration of Eq. ( 1) by parts for r 1 gives where ρ (r 1 , ω) is the induced electric charge density [34]: Another integration of Eq. ( 1) by parts and the use of Eq. ( 2) give When the velocity of light is assumed to be infinite, i.e., c → ∞ , which ignores the retardation effect or indicates the quasi-static approximation, Eq. ( 5) becomes = φ ext (r, ω) .(6) In multiple metal nanostructures, the local electron density can be expressed by the sum of the local electron density n 0 (r 1 − R n ) of each metal nanostructure: where R n is the position of each nanostructure.When the coordinate origins are selected at each nanostructure position (i.e., r 1 = r ′ + R n ) and using Eq. ( 6), the following equation is derived in the quasi-static approximation where the dimension of the multiple nanostructure group is assumed to be much smaller than the wavelength of light: where ω p (r) = √ 4πe 2 n 0 (r) /m e is considered as the local electron density-dependent bulk plasmon frequency in each metal nanostructure because the bulk plasmon can be excited by the collective charge oscillation of the many electrons at the position r, which occupy the single electron energy levels as shown in Eq. ( 1), and the integral region of r ′ is confined to each nanostructure area.The bulk plasmon frequency ω p (r) assures that the bulk plasmon excitation disappears near the metal nanostructure surface where n 0 (r) ≈ 0 and outside the metal nanostructure where n 0 (r) = 0.

B. Localized plasmons in multiple metal nanospheres
Considering multiple metal nanospheres, the effective-, external scalar potentials and the Coulomb potentials can be expanded using the normalized spherical harmonics Y lm (Ω r ) with the complete orthonormal property as shown in Appendix B.
By substituting Eq. (B-1) into Eq.( 8) and assuming that electron density in each metal nanosphere has a step-function-like shape n 0 (r ′ ) = n 0 θ (a s − r ′ ) → dn 0 (r ′ ) /dr ′ = −n 0 δ (r ′ − a s ), whose model is useful to know the fundamental properties of the localized plasmons, the following equations for the radial components of the scalar potentials are derived: (1) Inside the metal nanosphere (0 ≤ r < a s ) http://www.sssj.org/ejssnt(J-Stage: http://www.jstage.jst.go.jp/browse/ejssnt/) e-Journal of Surface Science and Nanotechnology Volume 13 (2015) ( ) where a n is the radius of the nanosphere at R n , ω p = √ 4πe 2 n 0 /m e is the bulk plasmon frequency in the uniform electron gas, and is simply written as (2) Outside the metal nanosphere ( r ≥ a s ) The effective scalar potentials derived from Eqs. ( 9) and (10) coincide at r = a s .This indicates that the bulk plasmon term with 1 − ω 2 p /ω 2 in Eq. ( 9) disappears at the metal nanosphere surfaces by the surface plasmon shield-ing effect.
At the metal nanosphere surface (r = a s ), the following determinant derived by Eq. ( 9) or (10) gives the surface plasmon frequencies for multiple metal nanospheres: det where δ ll ′ is the Kronecker's delta and Ω r is the solid angle for r at |r| = a s .The first term of Eq. ( 11) gives the surface plasmon frequency ω = ω p √ l/(2l + 1) for the individual nanosphere and the second term gives the frequency shift caused by the interaction between the multiple metal nanospheres.The term is the structure factor for the surface plasmon excitation.Equation ( 11) is the formula used to calculate the surface plasmon frequencies in the multipole approximation for multiple metal nanospheres with different radiuses.
The surface plasmon frequencies for the nanosphere dimer are analytically calculated in the dipole approximation (l = l ′ = 1).When the system has rotational symmetry around the z axis, as shown in Fig. 1, indicating that only the terms m = m ′ = 0 contribute to the calculation, the following equation is derived from Eq. ( 10) at r = a s : ( ) After integration for the solid angle cos θ and ϕ in Eq. ( 11), the structure factor is given by Here, the integrand of Eq. ( 11) can be expressed by only cos θ terms, for example, Y 10 (Ω r ) ∝ cos θ, making analytical integration possible.The determinant of Eq. (11) Volume 13 (2015)

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gives the surface plasmon frequencies: The two kinds of frequencies correspond to the anti-bonding and bonding surface plasmon states in the nanosphere dimer at the bonding dipolar plasmon mode, as reported in other studies [7,9,11].From Eq. ( 12), the effective scalar potentials at the nanosphere surfaces are given by The poles of Eq. ( 15) give the two kinds of the surface plasmon frequency of Eq. ( 14).However, when a s = a n = a and φ 10;s ext (a s , ω) = φ 10;n ext (a n , ω), which is satisfied for the external electric field in the dipole approximation, the effective scalar potential is then given by The pole of Eq. ( 16) gives only the red-shifted surface plasmon frequency depending on the dimer spacing, whose property coincides with that of previous results [6][7][8][9][10][11].

C. Surface plasmon for periodic metal nanostructures
For periodic metal nanostructures, using Bloch's theorem, the effective scalar potential at R n can be expressed as where k is the wave vector of the surface plasmon.Substitution of Eq. ( 17) into Eq.( 8) in the quasi-static approximation gives The quasi-static approximation may not be valid for periodic metal nanostructures because the dimension of the periodic nanostructures becomes larger than the wavelength of light.However, the surface plasmon interaction range between the nanostructures is approximately expressed as 1/ |R n − R s | l+1 : l ≥ 1 and Eq. ( 18) is consid-ered to give proper results because the interaction range is confined to an area that is smaller than the wavelength of light.When the effective-, external scalar potentials and the Coulomb potentials are expanded by the normalized spherical harmonics using Eq.(B-1) and the electron density is expressed as n 0 (r ′ ) = n 0 θ (a − r ′ ), the following equation is derived outside the metal nanospheres: At r = a, the following determinant gives the dispersion relation of the surface plasmon: det Equation ( 20) is the formula used to calculate the dispersion relation of the surface plasmon frequencies for the periodic metal nanospheres.The surface plasmon frequencies for the nanosphere chain are analytically calculated in the dipole approximation (l = l ′ = 1).When the nanospheres are aligned on the z axis with equal spacing R, as shown in Fig. 2, where the terms of m = m ′ = 0 only contribute to the calculation, using Eq. ( 13), the structure factor B 10,10 (k, a) in Eq. ( 20) is given by where |R n − R s | = nR and k z is the wavenumber in the z direction.Then, Eq. ( 20) gives the dispersion relation of the surface plasmon: ] . ( Figure 3 shows the numerical result of f plasmon cannot be excited for this chain structure at the exact bonding state.

D. Electromagnetic fields and light emission from multiple dipole moments
The Maxwell equations for the electromagnetic fields E (r, ω) and B (r, ω) can be represented by the scalar potential φ eff (r, ω) and the vector potential A (r, ω) .When the Lorentz gauge and Gaussian units are used, the electromagnetic fields in the ω expression are given by with the Lorentz condition ∇ • A (r, ω) − i ω c φ eff (r, ω) = 0.A previous study [31] showed that the electric displacement D (r, ω) can be expressed using the local electron density-dependent dielectric function ε (r, ω): Substituting Eq. ( 23) into the Maxwell equation, the following equation is derived:

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where j (r, ω) is the current density induced by the plasmon excitations [34].The retarded induced vector potential outside the metal nanostructures is then given by because ε (r, ω) = 1.The induced scalar potential is also given by where G 0 (r − r 1 , ω) and ρ (r 1 , ω) are given by Eqs. ( 2) and ( 4), respectively.
Using the induced electric charge density in Eq. ( 4) and integrating by parts, the electric dipole moment p (ω), its first-order derivative ṗ (ω), and its second-order deriva-tive p (ω) with respect to time are given by In the multiple metal nanostructures, the local electron density can be expressed by Eq. ( 7) and the electric dipole moment of each metal nanostructure at R n is defined as where the integral region of r ′ is confined to each nanostructure area.Then, using the calculations shown in Appendix C, the induced scalar and vector potentials are given in the electric dipole approximation by where is the unit vector as shown in Fig. 4. Substitution of Eqs. ( 30) and ( 31) into Eq.( 23) gives where only the phase differences between the multiple nanostructures are considered to clarify the interference effect for the electromagnetic fields.Using the Poynting vector S (r, t) = cE far ind (r, t) × H far ind (r, t) /4π, the light emission intensity or the photon number with energy hω per unit solid angle Ω : I ph (Ω, ω) is given in the time t http://www.sssj.org/ejssnt(J-Stage: http://www.jstage.jst.go.jp/browse/ejssnt/) e-Journal of Surface Science and Nanotechnology Volume 13 (2015) interval from −∞ to ∞ at r → ∞ by [34] Only the electromagnetic far-fields (1/r) contribute to the light emission.Equation (37) gives the light emission intensity from multiple metal nanostructures in the electric dipole approximation.Using Eq. ( 37), the light emission intensities from the metal nanosphere dimer in Fig. 1 and the metal nanosphere chain in Fig. 2 were analytically calculated by expanding the effective scalar potentials using Eq.(B-1).

Light emission intensity from metal nanosphere dimer
When the nanosphere dimer is aligned on the z axis as shown in Fig. 1, the system has rotational symmetry around the z axis, indicating that only the terms m = 0 and l = 1 contribute to the dipole moment [34].Then, the following result is obtained: ] In the calculation, only the z component of the gradient ∇ r ′ with the cos θ ′ ∂ ∂r ′ term contributes to the dipole moment, where the relation P 0 1 (cos θ ′ ) = cos θ ′ and the orthogonal relation for P 0 l (cos θ ′ ) were used [34].ε is a small positive value.
When the radiuses of the nanospheres are equal and the electron density is expressed as n 0 (r ′ ) = n 0 θ (a − r ′ ), the dipole moment of Eq. ( 38) is given by pn and from Eq. ( 36) ) 1 2 a 2 φ 10;n eff (a, ω) ) 1 2 a 2 φ 10;n eff (a, ω), (40) because the case ωe r • (R s − R n ) /c << 1 is considered.The effective scalar potential inside the nanosphere is given by where B 10,10 (R n − R s , r) is given by Eq. ( 13).It should be noted that the second bulk plasmon term in Eq. ( 41) disappears at r = a and the effective scalar potential coincides with Eq. ( 16).This indicates that only the surface plasmon excitation contributes to the electric dipole moments by the coupling between the bulk and surface plasmons.
When an external electric field E (r, ω) = e z E 0 is incident on the nanosphere dimer, indicating white light irradiation in the dipole approximation, the external scalar potential is given by Substituting Eqs. ( 40), (41), and ( 42) at r = a into Eq.( 37) gives the following light emission intensity from the metal nanosphere dimer: where θ is the angle between e r and e z , and Γ is the finite damping frequency caused by the finite lifetime of the single electron in the metal nanostructures.To correctly consider the damping, (ω + iη) 2 is replaced by ω (ω + iΓ) [37][38][39].The resonant frequency of the light emission is redhttp://www.sssj.org/ejssnt(J-Stage: http://www.jstage.jst.go.jp/browse/ejssnt/) Volume 13 (2015) Ichikawa shifted depending on the dimer spacing and the intensity becomes four times greater than that from a single metal nanosphere [34].
Equation ( 43) cannot be applied to the charge transfer plasmon mode where the local electron densities in both metal nanospheres change because of spill-out and tunneling effects near their surfaces [7][8][9][10][11].These effects decrease the surface plasmon frequency and the local electric field strength between the nanospheres.The surface plasmon-related terms in Eqs. ( 8) and ( 29) include the gradient of local electron density ∇ r ′ n 0 (r ′ ).The gradient of the electron density decreases near the surfaces by spill-out and tunneling effects, resulting in a decrease of the surface plasmon frequency and the local electric field strength.These effects should be further investigated in future study.

Light emission intensity from metal nanosphere chain
For periodic metal nanostructures, the dipole moment at R n in Eq. ( 29) is expressed using Bloch's theorem of Eq. ( 17) as Considering the case where the metal nanospheres have the same radius a and the step-function-like electron den-sity is aligned on the z axis with equal spacing R, as shown in Fig. 2, using Eq.(B-1), the dipole moment of Eq. ( 44) is given by pn Equation (36) gives where k z is the z component of the wave vector k, B 10;10 (k z , a) is given by Eq. ( 21), and an external electric field E (r, ω) = e z E 0 is incident on the nanosphere chain.
Summing n in Eq. ( 46) gives where N is the number of the nanospheres in the chain.Then, the light emission intensity is approximately given by where the case of ωR cos θ/c << 1 is considered.The result of Eq. (48) indicates that surface plasmon resonance does not occur at the condition where discussed in Section II.C.

E. Retardation effect on localized surface plasmon
For multiple metal nanostructures, Eq. ( 5) is given by The fourth term of the left-hand side of Eq. ( 49) is caused by the retardation effect on the bulk plasmon excitation.G 0 (r − r ′ + R s − R n , ω) is the structural Green's function, and is given at the condition [40,41] where the term C lm;l ′ m ′ ;l ′′ m ′′ , which is related to the Gaunt coefficient [40,41], is given by ) is known as the KKR structure factor [40,41] and only depends on the metal nanostructure arrangements.For the condition R s = R n , r > r ′ , the retarded Green's function is given by Eq. (B-2).
When multiple metal nanospheres with step-function-like electron density are considered, substitution of Eqs. ( 50) and (B-2) into Eq.( 49) without the fourth term gives where the case r ≥ a s , r > r ′ , ωa s /c and ωa n /c << 1 was considered and the approximated formulas in Eq. (B-3) were used.The fourth term of the left-hand side of Eq. ( 49) was ignored in Eq. ( 51) because this is in the order of ω 2 p a 2 s k 2 /ω 2 << 1 (k is the wavenumber of light), which is much smaller than the other terms.At the metal nanosphere surface r = a s in Eq. ( 51), the following determinant gives the surface plasmon frequency: det For the periodic metal nanospheres at r = a, the following determinant gives the dispersion relation of the surface plasmon frequencies using Bloch's theorem: det where the structure factor D lm;l ′ m ′ (k, ω) is given by The first terms in Eqs. ( 52) and (53) give the surface plasmon frequencies for the individual nanosphere, and the second terms give the frequency shifts caused by the interactions between the metal nanospheres considering the retardation effects on the scalar potentials.Compared with Eqs.(11) and (20) derived in the quasi-static approximation, Eqs. ( 52) and ( 53) have the advantage that the structure factors can be calculated by only summations without integrations for the solid angle, but they have the disadvantage that they include the frequency ω, making the calculation of the surface plasmon frequencies complicated.When the light velocity becomes infinite (c → ∞), Eqs. ( 52) and (53) might coincide with Eqs.(11) and (20), respectively.The retardation effects should be further investigated in future study.

F. Comparison with other numerical calculation methods
Several methods to numerically calculate the electromagnetic fields from multiple metal nanostructures have been developed such as the discrete dipole approximation (DDA) method [42,43], the boundary element method (BEM) [30,44], the finite-difference time-domain (FDTD) method [45,46].
The formulas in the dipole approximation of Eqs. ( 32) and ( 33) are similar to those derived by the DDA method.In this method, the local electric fields E loc (r n ) from multiple dipole moments p n are self-consistently calculated by using the dipole polarizabilities α n with the relation of p n = α n E loc (r n ).In our method, the local electric fields are calculated by using the local dipole moments in Eq. (29) where the coupling between the localized bulk and surface plasmons is properly considered by the effective scalar potentials and the gradient of the local electron density as shown in Eq. ( 8).These are not considered in the DDA method where the classical dipole polarizabilities are used.
The BEM has been also widely used to calculate the electromagnetic fields from multiple metal nanostructures [30,42].In the BEM, the induced scalar and vector potentials are used to calculate the electromagnetic fields from multiple metal nanostructures, which is similar to Volume 13 (2015) Ichikawa our method.However the classical Drude dielectric function and the induced surface (or interface) charge density are used for the calculations in the BEM, which cannot properly consider the coupling between the localized bulk and surface plasmons in metal nanostructures.In our method, the induced surface and bulk charge densities by the plasmons are properly considered by only using Eq. ( 4) The FDTD method has been most widely applied to calculate the electromagnetic fields for metal nanostructures by solving the Maxwell equations that are discretized on the Yee lattice points.This can be applied to metal nanostructures with arbitrary shapes.In these methods, however, the classical Drude dielectric functions are also used for metal nanostructures, which cannot properly consider the coupling between the localized bulk and surface plasmons.The three-dimensional electromagnetic are calculated in this method.On the other hand, the electromagnetic fields can be calculated by only using the one-dimensional scalar potential and the gradient of the local electron density by our method, e.g.Eq. ( 29).Analytical results can be also obtained in some cases.These are advantages of our method.However in order to calculate the electromagnetic fields for multiple metal nanostructures with arbitrary shapes, the integral equation such as Eq. ( 8) has to be solved.This is the disadvantage of our method.

III. CONCLUSION
A theory of localized plasmons for multiple metal nanostructures was derived using the RPA at the high frequency condition by developing previous theories.Selfconsistent integral equations were derived to calculate the localized plasmons frequencies for multiple metal nanostructures with non-periodic and periodic arrangements.Analytical calculations of a metal nanosphere dimer and chain with step-function-like electron density in the dipole approximation showed red-or blue-shifts of the LSP frequencies depending on the spacing between the metal nanospheres.
The light emission phenomena of the LSPs were also investigated for the multiple metal nanostructures.The dipole moments in multiple metal nanostructures were calculated to obtain their electromagnetic fields and the light emission intensity.Analytical calculations of a metal nanosphere dimer and chain with step-function-like electron density showed red-shifted light emission near the surface plasmon frequency condition and no resonant light emission under some conditions for the metal nanosphere chain.
The retardation effect on the LSPs was then investigated for multiple metal nanostructures by applying the structural Green's function method, which is used to calculate the electronic structures of condensed matter.The determinants were derived to calculate the surface plasmon frequencies for multiple metal nanospheres, which considered the retardation effects between the metal nanospheres.However, the relationship between the treatments derived by the quasi-static approximation and the structural Green's function method still remain a problem that needs to be solved.These results show that the theory presented here is useful to study the localized plasmons for multiple metal nanostructures.Considering an effective scalar potential φ eff (r 2 , ω) within the electron gas in metal nanostructures, an electron charge density induced by the effective scalar potential is given in the frequency expression by [12,31] ρ ind (r 1 , ω) =

∫
dr 2 e 2 P 0 (r 1 , r 2 , ω)φ eff (r 2 , ω). (A1) The polarization of the non-interacting electron gas P 0 (r 1 , r 2 , ω) is given by where the infinitesimal imaginary term iη is required by the causality of the one-particle Green's function for a single electron.where G 0 (r − r 1 , ω) is the retarded Green's function of Eq. (2).When an external scalar potential φ ext (r, ω) is considered, the retarded effective scalar potential satisfies