Letter pubs.acs.org/NanoLett
Quantum Plasmonics: Optical Properties of a Nanomatryushka Vikram Kulkarni,† Emil Prodan,‡ and Peter Nordlander*,† †
Department of Physics and Astronomy, MS 61, Rice University, Houston, Texas 77005, United States Department of Physics, Yeshiva University, New York, New York 10016, United States
‡
S Supporting Information *
ABSTRACT: Quantum mechanical effects can significantly reduce the plasmon-induced field enhancements around nanoparticles. Here we present a quantum mechanical investigation of the plasmon resonances in a nanomatryushka, which is a concentric core−shell nanoparticle consisting of a solid metallic core encapsulated in a thin metallic shell. We compute the optical response using the time-dependent density functional theory and compare the results with predictions based on the classical electromagnetic theory. We find strong quantum mechanical effects for core−shell spacings below 5 Å, a regime where both the absorption cross section and the local field enhancements differ significantly from the classical predictions. We also show that the workfunction of the metal is a crucial parameter determining the onset and magnitude of quantum effects. For metals with lower workfunctions such as aluminum, the quantum effects are found to be significantly more pronounced than for a noble metal such as gold. KEYWORDS: Time-dependent density functional theory (TDDFT), localized surface plasmon resonance (LSPR), quantum plasmonics
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Recently, there has been a strong interest in multiconcentric nanoshells known as “nanomatryushkas” (NMs).37−48 These structures are built around a solid metallic core, with additional concentric metallic, semiconducting, or dielectric shells added layer by layer (see Figure 1A). Due to the strong coupling of the plasmons on each of the metal surfaces, these structures display an unusually large tunability factor. An additional advantage of NMs over the conventional metallic nanoshells with a silica core is that they can be fabricated in very small sizes: the solid metallic particles serving as cores can be synthesized in much smaller sizes compared with the silica cores. A recent advance in DNA based synthesis of NMs has enabled the experimental fabrication of NMs with subnanometer separations between adjacent metallic surfaces.49 These particles were shown to generate strong Raman scattering signals from molecules in the gap, making them ideal for surface enhanced spectroscopy applications and as Raman tags. In addition due to their small size and high tunability, these particles are likely to be very useful in biomedical applications. However, as the intermetallic gaps become narrower and reach sub nanometer widths, QM effects are likely to become important. In this paper, we present a full QM investigation of the optical response of NMs and show that, for core−shell spacings smaller than 5 Å, the classical electrodynamics description fails and a QM treatment is needed for a proper modeling of their
oble-metal nanoparticles are key components in a wide range of applications spanning from surface-enhanced spectroscopies to medical applications.1−3 These applications are based on the tunable plasmon response of the nanostructures and on the large plasmon-enhanced local electric fields at their surfaces. The plasmon resonances, which are highly dependent on both chemical composition and geometry, can be tuned to wavelengths spanning the visible and near-infrared regions of the electromagnetic spectrum. Examples of nanostructures that have demonstrated plasmon tunability and significant field enhancements include spherical core−shell particles such as nanoshells,4,5 nanorods,6−10 nanorings,11−17 and nanoparticle dimers.18−23 Much of the recent research in plasmonic nanostructure fabrication is focused on developing strategies for synthesis of plasmonic nanostructures with narrow junction gaps or sharp protrusions. This trend is primarily guided by the classical electromagnetic theory, which predicts strong electric field enhancements around such structural features. However, for subnanometer junction gaps, quantum mechanical (QM) effects can strongly screen and reduce the electric field enhancements.24 One such QM effect is electron tunneling across the gaps. Another QM effect, which also directly influences the field enhancements, is the nonlocal screening, that is, the spread of the plasmon-induced surface charges, which in a QM treatment are no longer strictly localized on the surface but instead distributed in a volume around the surface.25 While such effects can be modeled empirically using classical nonlocal dielectric functions,22,26−32 the accuracy of such empirical methods can only be established by comparing with full QM calculations.24,25,33−36 © 2013 American Chemical Society
Received: July 18, 2013 Revised: October 30, 2013 Published: November 8, 2013 5873
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Figure 1. (A) Geometry of a nanomatryushka. (B) Absorption cross section calculated using classical electromagnetic theory for a NM of geometry (R1, R2, R3) = (16, 18, 30) Å. Left (right) inset represents induced electric field at 1.81 (2.87) eV. (C) Plasmon hybridization diagram. “NC” = nanocore. “NS” = nanoshell. “NB” = nonbonding. “NM” = nanomatryushka. (D) Schematic surface charge distributions for the 1.81 eV (left) and 2.87 eV (right) modes.
enhancements will also remain the same after such scaling. In Figure 1B, the optical spectrum is seen to display two localized surface plasmon resonances (LSPRs) at approximately 1.81 and 2.87 eV. The left and right insets show the induced electric field amplitudes corresponding to the two resonances. Both LSPRs yield a strong electric field in the gap region, which reiterates the potential of these structures as SERS substrates. The low energy mode only produces a strong field in the gap, while the higher energy mode produces a strong enhancement outside the R3 surface as well. For the particular aspect ratio chosen in these calculations, the higher energy resonant mode has a larger absorption cross section compared with the lower energy resonant mode. The peak heights are determined by the magnitudes of the induced dipole moments, which are extremely sensitive to the NM geometry. An intuitive physical understanding of the origins of the two resonant modes seen in Figure 1B is obtained from the plasmon hybridization (PH)4 diagram in Figure 1C. The starting point for this diagram is the building blocks of the NM: the solid spherical nanocore and the outer nanoshell. These two nanostructures, when completely separated from each other, support a resonant core dipole mode denoted by |ω−>NC and a bonding and antibonding nanoshell dipole mode |ω±>NS as illustrated in Figure 1C. When the building blocks are assembled together in the NM geometry, these resonant modes interact and hybridize as illustrated by the midcolumn of PH diagram. For this particular NM geometry, the 1.81 eV resonant dipole mode, denoted as |ω−−>NM in the PH diagram, is a bonding hybridized resonance formed primarily from the nanocore plasmon mode |ω−>NC and the bonding shell plasmon mode |ω−>NS. The higher energy 2.87 eV resonant dipole mode, denoted as |ω−+>NM in the PH diagram, is a bonding mode formed from the nanocore plasmon mode |ω−>NC and the antibonding shell plasmon |ω+>NS, with a slight admixture of the bonding shell plasmon |ω−>NS. A third mode |ω−+>NB is also predicted as an antibonding combination of the sphere resonant mode |ω−>NC and the antibonding resonant
optical response. Due to the spherical symmetry of the particle, we are able to model realistic size nanoparticles where the plasmon energies are fully developed and quantum-size effects absent. We also investigate the effect of the workfunction on their optical response and show that QM effects are significantly enhanced for low workfunction metal such as aluminum, when compared to gold. Classical Modeling. Figure 1A shows the geometry of a nanomatryushka, together with the structural parameters (R1, R2, R3) representing the radius of the inner metallic sphere and the inner and outer radii of the metallic shell. The Angstrom (Å) unit will be used for distance throughout the paper. To avoid unnecessary complications, the region between the sphere and the shell is assumed to be vacuum, and the nanostructure is embedded in vacuum as well. The metal is modeled as a jellium with a Wigner−Seitz radius of rs = 3.02 Bohr (1.6 Å), a background polarizability of ϵ∞ = 8.0, and a damping parameter γ = 0.135 eV. These parameters correspond to the jellium model for Au, which in the infrared frequencies displays a Drude-like dielectric permittivity function: ϵ(ω) = ϵ∞−
ωB2 ω(ω + iγ )
(1)
with a bulk plasmon frequency of ωB = 9.07 eV. Our damping parameter γ is smaller than the experimental value for Au to display the plasmon resonances more clearly. Figure 1B shows a generic normalized classical absorption cross section of a NM. Since for the moment we want to discuss only the qualitative features of the optical response, a classical analytical quasistatic approach50 was used in these calculations. As it is well-known, in the electrostatic limit, the optical properties are scale invariant, that is, if R1, R2, and R3 are all multiplied by the same factor λ, the only effect will be an overall scaling of the absorption cross section curves. The resonant frequencies and the relative intensities of the plasmon peaks will remain the same. The relative electric field 5874
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mode of the shell |ω+>NS. This third resonant mode is not visible in the spectrum shown in Figure 1B because of its low dipole moment. An illustration of the surface charge configurations of the two visible resonant modes is shown in Figure 1D. Based on the diagrams in Figure 1, we predict that the 1.81 eV resonance will be particularly prone to quantum effects, because of its strong electric field amplitude in the cavity between the core and the shell. Quantum Mechanical Modeling. For the QM calculations, we have extended our previously developed algorithms,33,34 which have been used extensively in the past to study the optical response of the spherical nanoshells. The algorithms are based on the time-dependent density functional theory,57−59 in the adiabatic local density approximation (TDLDA).60,61 As already indicated, the metallic gold phases are modeled within the jellium approximation, where the inclusion of a background ion-core polarizability and a pseudopotential for the core electrons ensures that the right workfunction (5.4 eV) is obtained. We first calculate the electronic structure of the NMs, and then we use the linear response within the random phase approximation (RPA) to compute their optical properties. We take full advantage of the spherical symmetry of the system, which enables us to perform full quantum mechanical simulations of NMs with more than a million electrons, thus reaching the regime where the plasmon resonances are fully developed. Interesting details appear already in the electronic structure of NMs. Figure 2 compares the ground state electron-density distribution and the effective one-electron potential for two
NMs with different core−shell gaps. As one can see, for the NM with a larger gap (5.3 Å), the electrons are mostly confined to the inner sphere and outer shell, with negligible electron presence in the gap region. At the same time, the effective oneelectron potential exhibits a significant potential barrier in the gap. For the NM with a smaller gap (1.1 Å), the potential barrier between the core and the shell is strongly reduced, allowing significant electron-density spill-out inside the gap. As we shall see, these features play an essential role in the optical properties of NMs. We emphasize that in a classical model that neglects tunneling, the electron presence will be strictly prohibited inside the gap, for all core−shell spacings. Figure 3 reports the normalized TDLDA absorption cross section spectrum for five gold NMs of geometries λ × (8.5, 9.5,
Figure 3. Normalized TDLDA absorption cross section for gold NM of dimensions λ × (8.5, 9.5, 15.9) Å. The scaling parameter λ is varied from 1 (bottom curve) to 5 (top curve). Inset is a zoom-in around the low energy resonance. Black arrows indicate the peak positions obtained in a classical simulation.
15.9) Å, with λ taking integer values from 1 to 5. The core− shell gap of these NMs varies from 1 to 5 Å. As we already mentioned, in a classical simulation the normalized spectra of these five NMs would be identical because of the scale invariance (the retardation effect can be entirely neglected for these NM sizes). The positions of the classical plasmon peaks are indicated by the arrows. The TDLDA results for the larger NMs (λ = 4 and 5) are seen to be in excellent agreement with the classical prediction, but there are dramatic differences when the core−shell gap is reduced. The most notable difference is the disappearance of the 1.81 eV resonance for smaller λ. A zoom-in on the spectra around this energy (see the inset of Figure 3) reveals that we are indeed seeing a true disappearance of the resonance and not just a mere attenuation of the plasmon peak. There are two possible explanations for this phenomenon: (1) the quantum size effect and/or (2) the electron tunneling between the core and the shell. The NM corresponding to λ = 1 contains less than a thousand electrons and may lack the continuum of conduction electrons energies required for a fully developed plasmonic response. However, we have also performed TDLDA simulations on NMs of larger size containing far more electrons, and there we observed the lower energy plasmon vanishing in a similar fashion to the results in Figure 3. (See Supporting Information for more details.) Therefore scenario (1) can be ruled out. Thus, we conclude that the disappearance of the low energy plasmon peak is due to the electron tunneling, which causes a charge transfer between the core and the shell. This process essentially short-circuits the core and the shell, resulting in a plasmonic response characteristic of a homogeneous particle.
Figure 2. Calculated conduction electron density (solid black) and effective l = 0 one-electron potential (solid blue) for a (15.9, 21.2, 31.7) Å NM (upper panel) and a (15.9, 16.9, 26.5) Å NM (lower panel). The Fermi energy is represented by the horizontal blue dashed lines and the jellium background with the black dashed lines. 5875
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In addition to the disappearance of the lower plasmon peaks, the QM calculations reveal a blue shift of the main plasmon peak at 2.87 eV, as the core−shell gap decreases. Such a blue shift was also seen in the previous investigation of nanoparticle dimers with sub nanometer gaps.24 Physically, the effect is due to nonlocal screening: the plasmon-induced charges on the core and inner shell surface are smeared out over an approximately 5 Å region around the surface. This smearing leads to a reduction in the Coulomb coupling between the core and shell plasmons and consequently a blue shift of the hybridized resonance. Figure 4 shows maps of the amplitudes of the induced electric fields at the resonant energies, for the five NMs
Figure 5. Comparisons of ground state electron density, CD (top), and (l = 0) one-electron effective potential (bottom) for a NM of dimensions (15.9, 16.9, 26.5) Å made of Au (red) and Al (blue). The black dashed lines show the jellium potential, and the horizontal dashed lines are the Fermi levels.
Figure 6 shows the TDLDA optical absorption spectra after the computations reported in Figure 3 were repeated for Al. Figure 4. Induced electric field distributions for the 1.81 eV mode (top) and the 2.87 eV mode (bottom) for the scaled NMs (λ = 1, 2, 3, 4, and 5) shown in Figure 3.
investigated in Figure 3. For λ = 5, the classical and the quantum field amplitudes distributions are almost identical: The low energy mode is seen to induce a strong field in the core−shell gap, and the higher energy mode to induce a strong field both in the gap and outside the shell. Little is changed for the NM with λ = 4, but as λ is further decreased, one can see a gradual attenuation of the electric field inside the gap region. For λ = 1, the charge transfer between the core and shell is so large that the system essentially behaves as a solid sphere, and the electric field is virtually zero in the core−shell gap. In the same time, the field amplitude map at the 2.87 eV resonance for the λ = 1 NM resembles that of a solid sphere rather than that of a core−shell structure. Further insight into the role of quantum effects can be gained by investigating the quantum plasmonic response as function of the workfunction of the metal. The simulations carried out so far have assumed a workfunction of 5.4 eV, appropriate for gold, but there are many other materials used in plasmonic applications with much lower workfunctions. For instance, aluminum, with a workfunction around 4.2 eV, has recently emerged as a novel material for plasmonic applications.53 Since a smaller workfunction results in a larger spill-out and a lower potential barrier inside the core−shell gap, one expects the QM effects to be more pronounced for lower workfunction metals. To quantify the effect of the workfunction, we have investigated the optical properties of aluminum NMs. The aluminum is modeled using a realistic Drude model with a Wigner−Seitz radius rs = 2.07 Bohr (1.1 Å), a background dielectric ϵ∞ = 1.25 and a workfunction of 4.2 eV. In Figure 5, we compare the ground state electronic structures of two NMs, one made of gold and the other made of aluminum. Due to the low Wigner−Seitz radius, aluminum possesses a larger conduction electron density. As one can clearly see, the Fermi energy lies significantly above the potential barrier for Al. It is thus expected that the optical properties for Al to exhibit stronger QM effects.
Figure 6. Normalized TDLDA absorption cross section for Aluminum NM of dimensions λ × (8.5, 9.5, 15.9) Å. The scaling parameter λ is varied from 1 (bottom curve) to 6(top curve). Inset is a zoom-in around the low energy resonance. Black arrows indicate the peak positions obtained from a classical simulation.
The inset of Figure 6, which zooms-in on the lower energy LSPR modes, looks noticeably different when compared with the inset of Figure 3: the λ = 3 (3 Å gap) mode is here significantly attenuated when compared to the 5.4 eV workfunction spectrum, where the LSPR appears as a clear peak. In the Supporting Information, we repeat the calculations for a metal defined by the same electronic parameters as was used for Au but with a workfunction 4 eV. Again, a similar trend emerges where the λ = 3 low energy mode is attenuated as compared to Au. Therefore the effects of tunneling are occurring at larger separations for lower workfunction metals. For Al, the low energy mode red shifts for higher λ before converging at λ = 5. This is in contrast to Au where the spectra are converged by λ = 4. Note there is a slight shift between the classical peak position and the quantum results even for large λ. This effect is due to the greater spill-out in Al, which changes the effective radii of the NM and thus red shifting the plasmon modes. Discussion. The recent observations that quantum mechanical effects can play an important role for the optical properties of strongly coupled plasmonic nanoparticles present 5876
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significant theoretical challenges. A full quantum mechanical description of the optical properties of realistic size nanoparticles is only possible for highly symmetric structures such as spheres, infinite cylinders, or semi-infinite thin films.33,34,51,52 Such systems are not of prime importance in SERS applications since the field enhancements for such structures are very modest compared to what is found, for instance, in nanoparticle junctions or near sharp structural features. In the quantum world, size matters: plasmon resonances are not fully developed in small systems where the metallic electronic structure is discrete rather than continuum-like. Small nanostructures consisting of a few hundred electrons typically display a quantum size effect both in their electronic density of states and in their optical response.25 Quantitative results from model calculations performed for small systems can therefore not always be extrapolated to larger, more realistic systems.54,55 The insight gained from fully quantum mechanical calculations for smaller model structures can be used to develop semirigorous approaches that can be applied to realistic systems. An example of such semiempirical approaches is the technique of nonlocal dielectric functions, which with the right parameters can describe quantum size effects and nonlocal surface charge screening within an otherwise classical electromagnetic description.22,26−28 Unfortunately the nonlocal dielectric function approach does not describe the very strong screening of the electric field enhancements in nanoparticle junctions caused by the QM effect of electron tunneling. For such systems, more elaborate semiempirical approaches are needed. Two recent examples of such approaches are the quantum corrected model (QCM)35 and the hybrid approach where the tunneling currents across a gap are evaluated using the Fowler−Nordheim field emission approach.56 More research on the development of accurate models for QM effects in realistic size systems is needed and likely to emerge in the near future. Quantum plasmonic effects are particularly important for the understanding of the origin of SERS enhancements. The observation that QM effects can reduce the field enhancements in nanoparticle junctions means that some previous classical calculations of the electric field enhancement factors may have been overestimated. This has important consequences for the interpretation of the enhancement factors in single molecule SERS experiments. With a smaller electromagnetic enhancement factor, the “chemical enhancement factor” may play a more important role than previously believed. Indeed, the finding of significant plasmon-induced electron transfer shows that electrons will be present in the junction in SERS experiments. It is not unlikely that such “tunneling electrons” can play a role in increasing the Raman cross sections for molecules in the junctions. Conclusions. We have conducted a first-principles investigation of the optical properties of subnanometer gap core− shell nanostructures known as nanomatryushkas. We have shown that substantial discrepancies exist between the quantum and classical models for both the absorption spectrum as well as the induced local electric fields. However, as the gap size is increased beyond 5 Å, the quantum predictions agree well with classical electromagnetic theory. The discrepancies are caused predominantly by electrons tunneling between the inner metal sphere and outer metal shell. Furthermore, we showed that the quantum effects are more important for low workfunction metals such as aluminum than for gold.
Letter
ASSOCIATED CONTENT
S Supporting Information *
Additional figures: (S1) Optical absorption spectra for larger nanomatryushkas calculated using TDDFT; (S2) Ground state properties for a fictitious system of workfunction 4 eV compared to Au NM; (S3) Optical absorption spectra for a fictitious system of workfunction 4 eV calculated using TDDFT. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors thank Drs. Jorge Zuloaga and Ke Zhao for valuable discussions. This research was in part supported by the Robert A. Welch Foundation under grant C-1222 and by the U.S. Army Research Laboratory and Office under contract/grant number W911NF-12-1-0407 and by NIH award NCRR S10RR02950 and an IBM Shared University Research (SUR) Award in partnership with CISCO, Qlogic and Adaptive Computing. E.P. acknowledges support from the U.S. NSF grants DMS-1066045 and DMR-1056168.
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