NANO LETTERS
Plasmon Hybridization in Nanoparticles near Metallic Surfaces
2004 Vol. 4, No. 11 2209-2213
P. Nordlander* Department of Physics and Astronomy, M.S. 61, Rice UniVersity, Houston Texas 77005-1892
E. Prodan Department of Physics, UniVersity of California, Santa Barbara, California 93106 Received August 26, 2004; Revised Manuscript Received September 22, 2004
ABSTRACT We apply the recently developed plasmon hybridization method to a solid nanosphere interacting with a metallic surface. We show that the plasmon energies of the nanoparticle exhibit strong shifts with nanoparticle−surface separation. Depending on the energy of the surface plasmon, nanoparticle plasmons can either red shift or blue shift with decreasing nanoparticle−surface separation. The shifts can be explained as resulting from image-like interactions with the metal surface and, more importantly, through hybridization between the nanoparticle plasmons and the delocalized surface plasmons of the substrate.
Plasmons in metallic nanostructures are of considerable interest in the field of nanoscience and technology. For instance, they provide the means of creating large local electromagnetic field magnifications, enhancing surface spectroscopies such as surface enhanced Raman spectroscopy (SERS) and surface enhance infrared absorption (SEIRA).1,2 The electromagnetic enhancements result from the surface charges associated with the plasmon oscillations and occur at frequencies near the plasmon energies of the substrate. To develop the right substrate for efficient sensing of a specific adsorbate, it is therefore crucial to understand which microscopic properties determine the substrate’s plasmon resonances. There is a wide range of nanostructures that can serve as substrates for enhanced surface spectroscopies. Nanoparticles adsorbed on conducting surfaces are of particular interest, both theoretically and experimentally.3-7 Due to the relatively complex geometry of the problem, theoretical investigations on the plasmonic properties of nanoparticles interacting with metallic substrates often involve relatively complicated algebraic expressions.8-13 Recently, we have developed a simple and intuitive description of the plasmon resonance in composite nanostructures, the plasmon hybridization concept.14 In this approach, the plasmon resonances of a complex nanostructure are expressed in terms of plasmon resonances of simpler constituent nanoparticles. The plasmon modes of the elementary nanoparticles hybridize to form composite plasmon modes in a manner analogous to how atomic orbitals interact 10.1021/nl0486160 CCC: $27.50 Published on Web 10/09/2004
© 2004 American Chemical Society
and form molecular orbitals. The method has been used successfully to describe the plasmon resonances in concentric spherical metallic nanoparticles15 and nanoparticle dimers.16 In this paper, we apply the plasmon hybridization method to calculate the plasmon energies of a nanoparticle near a metal surface and investigate their dependence on the nanoparticle-surface separation distance. We show that the plasmon energies of a nanoparticle can exhibit strong shifts as the particle approaches the surface. The shifts contain both image-like contributions, resulting from the interaction of the nanoparticle plasmons with their images in the metal, and hybridization shifts, resulting from the dynamic interaction between the plasmon modes. While the image interaction always results in a red shift of the plasmons with decreasing nanoparticle-surface separation, the effect of hybridization is either a red or blue shift, depending on the relative energies of the nanoparticle to the surface plasmons. In the plasmon hybridization method, the plasmon modes are expressed as incompressible deformations of the conduction electron liquid.14 For simplicity, we assume a uniform conduction electron density ns in the sphere and n0 in the surface. These electron densities correspond to bulk plasmon energies ωsB ) x4πnse2/me and ω0B ) x4πn0e2/me. The retardation effects are neglected and the interaction potentials between the plasmons are calculated using the instantaneous Coulomb potential.17 The incompressible deformations can be expressed in terms of a scalar potential η. In a semi-infinite metal
(z e 0), the most general expression of the deformation field is η)
dk B ∫P˙ Bk (t)eikB·Fb+kz (2π) 2
(1)
where the integration is over the parallel momentum B k) (k,φ′), b F ) (F,φ) denotes the lateral position along the surface, and z is the coordinate perpendicular to the surface. The quantity PBk is the amplitude of the surface plasmons. The deformation produces a surface charge
∫
σ(F b) ) n0e PBk (t)eikB·Fb
dk B (2π)2
(2)
where the integration is over the surface of the sphere. Because the system has azimuthal symmetry relative to the OZ axis passing through the center of the sphere, it is advantageous to expand the plane waves in eq 1 in azimuthal angle φ and to express the Lagrangian in terms of a Fourier transform of PBk, P(k,m) )
im x2π
∫ dφ′PBk exp[-imφ]
rather than in terms of PBk. The interaction takes the following form: yml xlRSlm∫ dkkP(k,m) exp[-kZ]Imlk ∑ lm
VI ) x2π An0nse2
(10)
which generates an electrostatic potential
∫
Φ(F b, z) ) 2πn0e PBk (t)eikB·Fb-k|z|
n0me 1 η∇ B η dS BσΦ dS 2 2
∫
where
dk B (2π)2
The Lagrangian describing the plasmon dynamics, L0 )
∫
(3) Imkl ) 15
n 0m e dk B k[P˙ Bk 2 - ωsp2PBk 2] 2 (2π)2
∫
(5)
nsme 2
∑l [S˙ 2lm - ωl2Slm2]
(6)
where Slm and ωl ) ωsB xl/(2l+1) are the amplitudes and energies of the sphere plasmons of angular momentum l. The surface charge generated by a single plasmon Slm on the sphere is given by σlm(Ω) ) nse
x
l SlmYlm(Ω) R3
(7)
where Ω ) (θ,φ) are the polar coordinates. We now bring the two systems together and consider the center of the solid metallic sphere at a finite distance Z (Z > R) above the metallic surface. The interaction between the sphere and flat surface plasmons is given by VI ) 2210
1
∑∫σlmΦ dS
2 lm
(11)
(4) yml )
(8)
x
2l + 1 (l - m)! 4π (l + m)!
is the normalization constant for the spherical harmonics. After rescaling the degrees of freedom in the Lagrangian,
where ωsp ) ω0B/x2 is the well-known energy of a flat surface plasmon. For a solid sphere of radius R, the plasmon dynamics is described by the following Lagrangian:15 Ls )
∫Jm(kR sin θ)Pml (cos θ)e-k R cos θsin θ dθ
Here, Jm and Pml are the cylindrical Bessel functions and Legendre polynomials, respectively and
takes the following simple form for the semi-infinite metal: L0 )
(9)
Slm f P(k,m) f
x x
2π k
2 S nsme lm 2 P(k,m) n0me
(12)
the total Lagrangian is written as L ) ∑mLm, where Lm has the following relatively compact form Lm )
∑l [S˙ 2lm - ωl2S2lm] + ∫ dk[P˙ 2(k, m) - ωsp2P2(k,m)] x2πω0BωsB∑yml xlRSlm∫ dk exp[-kZ]P(k,m)Imlk
(13)
l
Since there is no coupling between plasmon modes of different azimuthal symmetry, the equations of motion can be solved independently for each m. The integrals are replaced by discrete sums and the plasmon energies are obtained directly from the EulerLagrange equations. In the calculations presented above, we used 800 k-points from zero to 5 nm-1 and all sphere plasmons up to l ) 50.18 The radius of the sphere is assumed to be 10 nm and its electron density is chosen such that ωsB ) 9.06 eV, corresponding to gold. The calculations are fully converged for nanosphere-surface separations presented below. Nano Lett., Vol. 4, No. 11, 2004
Figure 1. Schematic illustrating the interaction of a nanosphere with a metallic surface. Panel (a) shows the image forces obtained assuming a perfect response of the surface for l ) 1 and l ) 2 sphere plasmon. The surface mediates an interaction between plasmons of different l, resulting in a distance-dependent hybridization of the nanosphere plasmons in a manner similar to the nanosphere dimer.16 Panel (b) illustrates the plasmon hybridization principle. The nanoparticle plasmons interact with the surface plasmons. This interaction results in shifts and an effective hybridization between nanoparticle plasmons of different angular momentum l.
The structure of the eigenvalue problem is illustrated in Figure 1b. The surface plasmons and nanosphere plasmons couple through the matrix element Imkl. Since the surface plasmons have a finite interaction with all sphere plasmons, they mediate an indirect interaction (hybridization) between sphere plasmons of different angular momentum l. We first calculate the energy shift of the plasmon resonances for a nanosphere near a surface of infinite plasmon energy. In this case, the metal surface responds instantaneously to the nanosphere plasmons. The results are shown in Figure 2. They are identical to what we would obtain from a simple image model of the surface (illustrated in Figure 1a) in which a nanosphere plasmon would interact both with its own image and with the images of the other nanosphere plasmons of the same azimuthal symmetry m. The picture is very similar to the formation of dimer plasmons from individual nanoparticle plasmons.16 At large distances, the red shift of the lowest energy l ) 1 nanosphere plasmon follows a Z-3 behavior. At closer distances, the red shift is faster because of the hybridization with nanosphere plasmons of larger angular momentum. Here and in the following, we refer to the nanosphere plasmons using their asymptotic angular momentum, although they will contain a significant admixture of other angular momenta at smaller separations. The figure also shows that the m ) ( 1 plasmons exhibit a weaker interaction with the surface than the perpendicular m ) 0 Nano Lett., Vol. 4, No. 11, 2004
Figure 2. The energy shift of the five lowest energy nanosphere plasmons as a function of separation Z from a perfect metal surface ω0B ) 900 eV. The lines used to draw the curves refer to the asymptotic (Z ) ∞) symmetry of the plasmon resonances and are l ) 1 (solid), l ) 2 (long dashed), l ) 3 (dashed), l ) 4 (short dashed), and l ) 5 (dotted). Panel (a) shows the m ) 0 plasmons (oriented perpendicular to the surface) and panel (b) shows the m ) ( 1 plasmons (orientated parallel to the surface).
plasmons. This anisotropy is simply caused by the mdependence of the Coulomb interaction between multipoles. We consider now the interaction between the same nanosphere and a surface of the same electron density, ω0B ) ωsB ) 9.06 eV. The energy shifts of the plasmon resonances are shown in Figure 3 as functions of nanoparticle-surface separation distance. Comparing Figure 2 and Figure 3, one can see that, even at larger distances, the red shifts are much larger for the present situation. This is because of the hybridization of the sphere plasmons with the surface plasmons. The surface plasmons are of higher energy (ωsp ) 6.4 eV) than all the sphere plasmons and approximately 1 eV above the energy of the lowest nanosphere plasmon (l ) 1). The small energy separation between the surface and sphere plasmons facilitates the hybridization. The effect is the same for all sphere plasmons because they are all of lower energy relative to the surface plasmons. This example clearly demonstrates that the simple image model of the surface can be inadequate and how it is essential to model the surface using the right surface plasmon frequency. In Figure 4, we plot the imaginary part of the dipole polarizability15 of the plasmon resonances in the system depicted in Figure 3 for several nanoparticle-surface separations. The upper panel (a) shows that for large separations, the optical absorption is characterized by a single peak located at the energy of the l ) 1 plasmon resonance. The 2211
Figure 3. The energy shifts of the five lowest energy nanosphere plasmon modes as a function of separation Z from a metal surface with the same bulk plasmon energy, ω0B ) ωsB ) 9.06 eV. Panel (a) shows the m ) 0 plasmons (oriented perpendicular to the surface) and panel (b) shows the m ) ( 1 plasmons (orientated parallel to the surface). The types of lines used in the plot are the same as in Figure 2.
weak feature at a frequency of 6.4 eV for Z ) 25 nm is due to the antibonding l ) 1 plasmon resulting from the interaction of the surface plasmons and the sphere plasmons. This mode is primarily composed of surface plasmon modes and has very little weight. Panel (b) shows the polarizability for intermediate distances. Again the optical response is determined by the plasmon resonances, which are now significantly red shifted compared to panel (a). For Z ) 15 nm, the l ) 2 plasmon becomes visible at 5.4 eV. This is caused by the finite admixture of the bare l ) 1 nanosphere plasmons. In panel (c), the dipole polarizability is shown for small Z. For Z ) 11 nm, four plasmon resonances become visible, corresponding to the l ) 1-4 plasmon resonances. The energy of the resonances in the calculated optical spectra are in perfect agreement with the results shown in Figure 3a. We now consider the case where the surface plasmon energy is lower than the lowest nanosphere plasmon energy. The results are shown in Figure 5, which displays the energy shifts of all plasmons in the system. The lowest levels derive from the surface plasmons. The degeneracy of these modes is lifted and a few states split off and red shift with decreasing nanoparticle-surface separation. The lowest energy states contain a small but finite admixture of the dipole active ω1 sphere plasmon, and may therefore be visible in a light scattering experiment. The upper curves derive from the nanosphere plasmons. In contradistinction with the situation discussed in Figure 3, it can be seen that hybridization shifts 2212
Figure 4. Imaginary part of the dipole polarizability for m ) 0 as a function of frequency for different nanoparticle separations Z. The system is the same as shown in Figure 3. Panel (a) shows Im[R] for Z ) 50 nm (dotted line) and Z ) 25 nm (solid line). Panel (b) is for Z ) 20 nm (dotted) and Z ) 15 nm (solid). Panel (c) is for Z ) 13 nm (dotted) and Z ) 11 nm (solid line). The spectra were calculated assuming a broadening of 100 meV. The same arbitrary units for the polarizability were used in all panels to illustrate the relative contributions of the different plasmon modes.
Figure 5. Energy shift of the m ) 0 sphere-surface plasmons as a function of sphere-surface separation when ω0B ) 6.0 eV and ωsB ) 9.06 eV. The types of lines used in the plot are the same as in Figure 2.
the nanosphere plasmons to higher energy with decreasing nanoparticle-surface separation. The admixture of surface plasmons at the smallest separations is relatively small, and Nano Lett., Vol. 4, No. 11, 2004
In conclusion, the plasmon hybridization method has been extended to the nanoparticle/surface system. The method provides a simple and intuitive explanation of the shift of the plasmon resonances as a function of nanoparticle-surface separation. The shift of the nanoparticle plasmons contains both image-like contributions and dynamical effects arising from hybridization of the sphere and surface plasmons. For surfaces with plasmon resonances above the nanoparticle plasmons, the plasmons red shift strongly with decreasing nanoparticle-surface separation. When the surface plasmons lie below the nanoparticle plasmons, a relatively weak blue shift results. Acknowledgment. This work was supported by the Multidisciplinary University Research Initiative of the Army Research Office, the Robert A. Welch Foundation under grant C-1222, and by the National Science Foundation under grants number EEC-0304097 and ECS-0421108. References
Figure 6. Energy shifts of the five lowest energy m ) 0 nanosphere-surface plasmons as a function of sphere-surface separation for metal surfaces of different bulk plasmon energies. Panel (a) shows the results for ω0B ) 6.0 eV and panel (b) shows the results for ω0B ) 2.0 eV. The types of lines used in the plot are the same as in Figure 2.
the l ) 1-3 modes all contain a relatively large admixture of the dipole active l ) 1 nanosphere mode. Figure 6 compares the example discussed in Figure 5 with the case where the nanosphere is near a surface of much lower electron density (ω0B ) 2.0 eV), such that the surface plasmon is of much lower energy than the nanoparticle plasmons. The figure only shows the modes that asymptotically connect to the individual nanosphere plasmons. The interaction between the nanoparticle and the surface decreases in the latter case, because of the detuning between the surface and sphere plasmons. The interaction between nanoparticle and surface plasmons is more complicated when the energy of the surface plasmon lies within the energies of the nanoparticle plasmon manifold. In this case, the intrananoparticle hybridization is disrupted and plasmons with different angular momenta can shift very differently depending on their energy relative to the surface plasmon. A detailed investigation of this effect along with an investigation of the plasmon hybridization of nanoparticles near thin metallic films is in progress.
Nano Lett., Vol. 4, No. 11, 2004
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