NANO LETTERS
Electronic Structure and Optical Properties of Gold Nanoshells
2003 Vol. 3, No. 10 1411-1415
E. Prodan, P. Nordlander,* and N. J. Halas Department of Physics and Rice Quantum Institute, Rice UniVersity, Houston, Texas 77251 Received August 1, 2003; Revised Manuscript Received August 29, 2003
ABSTRACT Using the TDLDA method, we investigate how the polarizability of the d electrons of the gold atoms influences the electronic and optical properties of metallic nanoshells. It is shown that a polarizable jellium background can introduce a significant shift of the plasmon resonances. The results of the study show that the theoretically calculated optical absorption spectra for gold nanoshells with a gold sulfide core are in excellent agreement with experimental data.
The optical properties of metallic nanoparticles have recently attracted considerable attention.1 A particularly interesting structure is the metallic nanoshell,2 which consists of a metallic layer grown over a solid dielectric core. The plasmon frequency, which determines the nanoshell optical properties, can be tuned over a wide spectral range by simply varying the ratio of the inner to outer diameter of the shell. The tunability of the optical properties of metallic nanoshells enables several important applications such as resonant photooxidation inhibitors,3 optical triggers for drug delivery implants,4,5 environmental sensors,6 and Raman sensors.7 In a series of papers, we have developed a theoretical method based on the time-dependent local density approximation (TDLDA) to calculate the electronic and optical properties of metallic nanoshells.8-10 The method has been optimized and applied to real-sized nanoshells, where the number of conduction electrons is typically larger than 105. The tunability of these nanoparticles has been verified by our TDLDA jellium calculations.11 The method has also been extended to examine the effects of a dielectric core and an embedding medium.12,13 These studies have shown that the calculated optical properties of the metallic nanoshells are in very good agreement with results obtained from classical Mie scattering when the metallic phase is modeled by a Drude dielectric function. The Drude model, however, neglects the contribution to the dielectric function from the polarizable metal cores. It is well known that it is crucial to employ a realistic dielectric function for the nanoparticle in order to achieve quantitative agreement between experimental data and classical Mie scattering.2,14 This is especially true for noble metals, where the polarizability of the d electrons is known to contribute significantly to the dielectric response in the optical frequency range.14 In this paper, we examine * Corresponding author. E-mail:
[email protected]. 10.1021/nl034594q CCC: $25.00 Published on Web 09/19/2003
© 2003 American Chemical Society
the role of the d electrons in the nanoshell optical response, further generalizing our TDLDA implementation to include a dielectric background arising from the polarizable gold ion cores. We show that the inclusion of these effects leads to quantitative agreement between the theoretically calculated and experimentally measured optical absorption of gold nanoshells. To describe a nanoshell of inner radius r1 and outer radius r2, we will use the notation (r1, r2). The conduction electrons of the nanoshell are assumed to be embedded in a nonuniform frequency-dependent dielectric with (r) ) C for r < r1, (r) ) J for r1 < r < r2, and (r) ) E for r2 < r. This parametrization accounts for the presence of a dielectric core, a polarizable jellium background (metal ion cores), and a dielectric embedding medium. The presence of the dielectric media can be incorporated into the ground-state LDA jellium calculations by replacing the Hartree potential,15 VH(r) )
∫
n(r′) - n0 db′ r |b r - b′| r
(1)
with
{
V˜ H(r) ) J - E C - J 1 VH(r) + VH(r2) + V (r ) for r < r1 C EJ JC H 1 J - E 1 for r1 < r < r2 VH(r) + V (r ) J E J H 2 1 V (r) for r2 < r E H (2)
so that the usual boundary conditions are satisfied.
by the screening charges including the excitation field. In the quasistatic limit, which is generally accepted to be valid for particles smaller than 40 nm,14,16 the electric potential φC must satisfy the Poisson equation with the appropriate boundary conditions: after the solution is included in eq 3 and using the notation V1(r, r′) ) r2, V2(r, r′) ) dV1/dr, and R(r, ω) ) r2δn(r, ω), the RPA equation becomes R(r, ω) )
Figure 1. Effect of different dielectric background media on the electronic structure of the (70, 100) au gold nanoshell. The upper panel (a) shows the electron charge density, and the lower panel (b) shows the Coulomb potential (upper curves) and the effective electron potential (lower curves) in the nanoshell. The solid line corresponds to a nanoshell with E ) J ) C ) 1, the dotted line corresponds to E ) C ) 1 and J ) 8, and the dashed line corresponds to E ) 2, J ) 8, and C ) 5. The dashed and the dotted lines in panel a coalesce.
The overall effects of this inclusion on the electronic structure are small, as illustrated in Figure 1 for a model nanoshell of geometry (70, 100) au. The inclusion of a dielectric embedding medium or a dielectric core has a relatively minor effect on the electronic structure, as has been discussed previously.12,13 The largest effect is caused by the inclusion of J, the jellium background. Its presence results in a lowering of the effective potential in the nanoshell. The lowering of the electron potential increases the work function of the nanoshell, which reduces the electron spillout. The presence of the dielectric media will have a large effect on the screening when the system is excited with an external electromagnetic field. The external field will polarize the dielectrics, and as a consequence, the electrons will screen not only the excitation field but also the electric field generated by the polarized dielectrics. To include the contribution from the induced charges in the TDLDA formalism, we start from the TDLDA expression of the screening charge induced by an external electric field E0e-iωt b ez, δn(b, r ω) )
r ω)[δVxc(b′, r ω) + φC(b′, r ω)] ∫ db′r Π(0)(b,r b′; (3)
where Π(0) represents the response function of the independent electrons. The quantity δVxc represents the variation of the exchange-correlation potential due to the screening charge, and φC represents the Coulomb potential generated 1412
4π 3
∫ dr′ Π˜ (0)1 (r, r′; ω) r′-2V′xc(n(r′)) R(r′, ω)
∫ dr′ ∫ dr′′ Π˜ (0)1 (r, r′; ω)
(4)
V1(r′, r′′) R(r′′, ω) (r′′)
V2(ri, r′) R(r′, ω) dr′ (r′)
4π
∑ X(i)(r, ω) ∫ 3 i)1,2
∫ dr′ Π˜ (0)1 (r, r′; ω) φ0C(r′)
where X(1)(r, ω) ) X(2)(r, ω) )
ξ - 1 1 C - J X1(r, ω) X2(r, ω) (5) ξ C + 2J ξ
()
ξ - 1 r2 3 1 E - J X1(r, ω) + X (r, ω) 2ξ r1 ξ J + 2E 2
and Xi(r, ω) ) r2i
∫ dr′ Π˜ (0)1 (r, r′; ω) V1(r′, ri)
(6)
In these equations, φ0C represents the unscreened response of the dielectrics to the electric field: φ0C(b) r ) (-r +
r2i σ(i) ∑ 0 V1(r, ri)) E0 cos θ i)1,2
(7)
where σ(1) 0 )
1 C - J 3E ξ C + 2J J + 2E
σ(2) 0 )1-
(8)
1 3E ξ J + 2E
are the classical polarization charges and ξ is just a dielectric factor,
()
C - J E - J r 1 ξ ) 1 - 2 C + 2J J + 2E r2
3
(9)
The notation Π ˜ 01(r, r′; ω) represents the l ) 1 component of the response function of the system in the independent electron approximation,17 multiplied by the factor r2r′2. The Nano Lett., Vol. 3, No. 10, 2003
sion for the sphere plasmon is
x
ωS(J) ) ωB
1 2 + J
Thus the cavity plasmon depends more sensitively on J than the sphere plasmon does, resulting in a large shift of the ω+ nanoshell plasmon. If we include all of the dielectric background media, then the symmetric mode is pushed further into the red while the antisymmetric mode remains at the value imposed by the dielectric constant inside the shell. The Figure also shows the position of the plasmon modes obtained using classical Mie scattering and a Drude dielectric function for the shell, (ω) ) J -
Figure 2. Effect of different dielectric background media on the optical absorbance of the (70, 100) au gold nanoshell. Panel a is for E ) 2, J ) 1, and C ) 5. Panel b is for E ) C ) 1 and J ) 8. Panel c is for E ) 2, J ) 8, and C ) 5. For comparison, we include (dashed line) the calculated optical absorbance for a nanoshell with E ) C ) J ) 1 in each panel. The arrows indicate the position of the resonances predicted by classical Mie theory and eq 11.
methods used for solving the RPA equations have been discussed previously.9,12 To understand the effect of different dielectrics qualitatively, we solve the modified equations for the test nanoshell shown in Figure 1 with the same dielectric constants. The results are shown in Figure 2. Each optical absorption spectrum is dominated by two dipolar plasmon modes. The low-energy (long-wavelength) mode ω- corresponds to a symmetric coupling of the cavity plasmon on the inner surface and the sphere surface plasmon on the outer surface. The higher-energy (shorter-wavelength) mode ω+ corresponds to antisymmetric coupling.11 The presence of the dielectric core and embedding medium (C ) 5 and E ) 2) introduces a red shift of both plasmon resonances as previously shown.13 This is clearly illustrated in panel a of Figure 2. The effect of a polarizable jellium background (J ) 8) also introduces a red shift of both plasmon resonances. The effect is strongest for the ω+ mode. To understand this, we invoke the simple cavity-sphere surface plasmon interaction model described in an earlier publication.12 The ω+ mode contains a larger admixture of the cavity plasmon than the ω- mode. For a cavity inside a uniform jellium with a background dielectric constant J, the energy of the cavity plasmon is
x
ωC(J) ) ωB
2 1 + 2J
For a uniform sphere in vacuum, the corresponding expresNano Lett., Vol. 3, No. 10, 2003
ω2B ω2 + iγω
(10)
where ωB is the bulk plasmon energy corresponding to rs ) 3, a value that is also used in our self-consistent jellium calculations. The plasmon frequency is independent of γ. As can be seen from the Figure, the agreement between the TDLDA calculations and classical Mie scattering results is excellent. There is a slight discrepancy for the ω+ mode. Given the large value of J and the fact that the resonance moved from approximately 150 to 385 nm, the 15-nm difference between the classical Mie predictions and our TDLDA calculation can be considered negligible. We now incorporate a realistic polarizability J and compare our results directly to experimental measurements. The polarizability of the jellium background J can be obtained by decomposing the experimentally measured bulk dielectric function into (ω) ) J(ω) + D(ω) - 1, where D(ω) ) (1 - ω2B)/(ω2 + iγω) is the Drude dielectric function of the conduction electrons in bulk. In this procedure, the width γ should be taken to be the same as the artificial width that will be used in the TDLDA calculations.18 The experimental gold nanoshells in this comparison have their plasmon resonances in the region far from interband transitions (λ > 600 nm). In this range of frequencies, the experimentally measured dielectric function for gold is very well approximated by an expression such as (ω) ) ∞ -
ω2B ω2 + iΓω
(11)
Depending on which experimental measurement is used, the best fit ranges from ∞ ) 10 (ref 19) to 8.20 A very recent experiment for wavelengths around 500 nm suggests that ∞ ) 8.7.21 The large value for gold is due to the completely filled d band of the cores. From this discussion, it is clear that J can be chosen as a constant and that the value should be close to ∞. In our self-consistent calculations, we found that J ) 8 gives us the best agreement with the experiment. The gold nanoshells have a Au2S core (C ) 5.4) and were suspended in water (E ) 1.78). These values of the dielectric media were used in the calculations presented here. 1413
Figure 3. Comparison between the calculated (circles) and experimentally measured (solid line) absorbance for three different gold nanoshells. Panel a is a nanoshell of dimension (4.1, 5.1) nm, panel b is for a nanoshell of dimension (8.6, 9.9) nm, and panel c is for a nanoshell of dimension (13.1, 14.9) nm. The dashed line is the absorbance of solid gold colloidal spheres that are also present in the solution. The parameters used in the TDLDA calculations are C ) 5.4, E ) 1.78, and J ) 8.
Figure 3 is a comparison between experimental data and our theoretical calculations of the optical absorbance. The experimental spectra display two peaks. The shortwavelength peak is the same in all three measurements and is caused by the presence of solid gold colloidal spheres with a plasmon resonance at 520 nm. The peak at longer wavelengths is the symmetric ω- plasmon mode of the gold nanoshells. Because of differences in the aspect ratio of the different nanoshells, there is a substantial difference in the position of the ω- plasmon modes for these particular nanoshells. The structures apparent in the calculated absorbance of the smallest nanoshell are due to the superposition of the collective resonance and single-electron excitations and will be further discussed below. Figure 3 clearly shows, however, that the calculated width of the plasmon resonances are far narrower than the experimental results. Several factors can contribute to the line shape of plasmon resonances.22 For a bulk gold system in the present frequency range, the dielectric background would contribute an imaginary part to J corresponding to Γ ≈ 0.07 eV in eq 11. Calculations including this term do not reveal any significant extra broadening of the plasmon resonances. A second factor contributing to the measured plasmon widths is the nonuniform size distribution. The experimental data shown in Figure 3 are snapshots of a solution of nanoshells taken during their growth. The experimental estimate of the size dispersion is a size distribution with a standard deviation of 11%. A third 1414
broadening mechanism in this particular experiment may be that the nanoshells are not perfectly spherical. It would be reasonable to expect that the nanoshell surfaces are relatively defect-rich during growth and may not be of uniform thickness. A fourth broadening factor is finite-size effects. When the dimension of the nanoparticle is smaller than the mean free path of the electrons, electron surface scattering will introduce an extra broadening of the plasmon resonance.14 This effect can be incorporated into the classical calculations by semiempirically modifying the Γ factor in the bulk dielectric function (eq 11). This modification influences the width of the resonance but leaves the energy unchanged. The mean free path effects are in principle included in the TDLDA calculations because the electronic structure was calculated for a finite system. However, because the energy levels in a finite nanoshell are discrete, the Landau damping of the plasmon resonances will depend sensitively on whether the plasmon energy is resonant with single-particle excitations.13 A detailed theoretical investigation of the finite-size dependence of the plasmon widths will be presented in a future publication. The fifth important factor contributing to the plasmon widths in nanoparticles is phase retardation effects. The phase retardation effects can be rigorously included only in the framework of current-density functional theory and will not be discussed here. For the present cases, they are expected to be negligible. We therefore investigate the effect of a finite-size distribution on the width of the plasmon resonance. As we have pointed out previously, our quantum treatment of the nanoshell will result in a much stronger size dependence of the optical properties than that obtained from the classical Mie scattering calculations13 because the energy of the ωplasmon is lower than the work function of the nanoshell and can be resonant with discrete single-electron excitations. As the geometry of the nanoshell is changed slightly, the plasmon resonances can move through these single-particle excitations and develop considerable structure. In Figure 4, we show the calculated absorbance of nine nanoshells with sizes (outer radius) within 11% of the nominal (4.1, 5.1) nm shell shown in panel a of Figure 3. The Figure shows that the interplay between the collective mode and the single-particle excitations can have a dramatic effect on the optical properties of each nanoshell. Some of the spectra show multiple-peak structure, and some are markedly asymmetric. In panel b, we show the optical absorbance of a Gaussian average of the optical absorbance of 15 different nanoshells of sizes within the 11% size distribution. The resulting width of the plasmon peak is in quantitative agreement with the experimental data. The various peaks that can be seen in the size-averaged optical absorbance are consequences of the finite sampling of the nanoshell size distribution. In conclusion, we have presented complete self-consistent jellium calculations of the electronic and optical properties of metallic nanoshells, including dielectric effects of the nanoshell core, the polarizable metal ion cores, and a polarizable embedding medium. In the case of noble metals, the effect of the polarizable metal ion cores results in a Nano Lett., Vol. 3, No. 10, 2003
in the outer radius induce strong changes in the plasmon line shape because of the interplay between the collective plasmon and discrete single-electron excitations. Acknowledgment. This work was supported by the Robert A. Welch foundation and by the Multi-University Research Initiative of the Army Research Office. References
Figure 4. Effect of size averaging on the line shape of the plasmon resonance. Panel a illustrates the absorbance for nanoshells of the same inner radius but different outer radii, uniformly distributed between (11% of the outer radius of the nanoshell shown in panel a of Figure 3. Panel b shows a Gaussian average (solid line) of the optical responses shown in panel a plus an additional six nanoshells. The dashed line represents a Lorentzian fit of the solid line, the triangles represent the experimental data, and the dotted line is the optical response of the solid gold particles that are also present in the solution.
substantial red shift of the symmetric nanoshell plasmon resonance. By choosing the dielectric constant of the jellium background to be similar to the experimental value, we were able to reproduce the experimentally measured absorption cross section of three nanoshells of very different dimensions. We found that the optical response of the nanoshells can be a very sensitive function of their geometry. Small variations
Nano Lett., Vol. 3, No. 10, 2003
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