Theory and Modeling of Plasmonic Structures - The Journal of

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Theory and Modeling of Plasmonic Structures Stephen K. Gray Center for Nanoscale Materials, Argonne National Laboratory, Argonne, Illinois 60439, United States ABSTRACT: Plasmonic structures, or systems generally containing nanostructured metallic components allowing for the exploitation of surface plasmon resonances, continue to draw much experimental and theoretical interest. This is due to the ability of surface plasmons to capture, concentrate, and propagate optical energy. This Feature Article discusses the basic theoretical principles and computational modeling of such structures. A variety of illustrations are also given, including optical transmission by nanohole arrays in thin metal films, remote-grating generation of narrow band plasmons, the excitation of dark modes in bipyramidal nanoparticles, optical transparencies in nanoparticle−quantum dot systems, and the size dependence of surface plasmon resonances in the limit of very small particle sizes.

1. INTRODUCTION Plasmonic structures, e.g., systems of metal nanoparticles, nanoholes, and slits in metal films, and hybrid systems also including dielectric layers, active media, or molecules are of much current interest. This is because when exposed to light surface plasmon (SP) excitations can occur, which can capture, concentrate, and propagate light energy.1−3 SPs arise from collective electronic excitations of conduction band electrons near metal surfaces, and they manifest themselves as evanescent electromagnetic surface waves. Depending on the structure and excitation mechanism, SPs may be stationary, often termed local surface plasmons (LSPs) or propagating, often termed surface plasmon polaritons (SPPs). In addition to being fundamentally interesting for their energy transduction properties, SPs are relevant to many applications including chemical and biological sensing,4 solar energy,5 optoelectronics,6 and even cancer treatment.7,8 The unbeknownst use of plasmonics goes back to Roman times with the fourth century Lycurgus cup, which was made with a recipe that incorporated trace amounts of gold and silver in the glass.9,10 When illuminated with white light from within its bowl, an observer outside sees red transmitted light. However, if light is incident on the cup from outside, then the scattered light seen by the observer is green in color. To a first approximation, the gold LSP resonance, where both scattered and absorbed light are enhanced near 530 nm, accounts for this behavior. (A more complete description would involve gold− silver alloy nanoparticles.9,10) The strong gold absorption near 530 nm, coupled with the general property that gold tends to absorb more strongly at lower wavelengths, leads to the observer seeing transmitted light with wavelengths larger than 530 nm that tends to be dominated by red light (≈650 nm). The “ruby red” color of medieval stained glass has a similar explanation. On the other hand, the scattered light intensity by the cup parallels the absorption and is dominated by © XXXX American Chemical Society

wavelengths closer to 530 nm, which is green light. It took until 1857 for the role of the gold nanoparticles to be deduced in such phenomena, and over 50 years later for a rigorous theory to be developed to explain such effects. Michael Faraday, in 1857, published a rambling but interesting account of his extensive experiments on colloidal gold, and a few quotes from this Bakerian Lecture11 reveal some of his key deductions: “The state of division of these particles must be extreme; ... As to the gold ruby in glass, I think a little consideration is sufficient to satisfy one that it is in the metallic condition. ... These considerations, with the sufficiency of the assigned cause to produce the ruby tint, are strong reasons, in the absence of any to the contrary, to induce the belief that finely-divided metallic gold is the source of the ruby colour.” In 1908, Gustav Mie’s analytical solution of the Maxwell equations for light interacting with a sphere of arbitrary refractive index and his detailed application of it to gold nanoparticles provided the theoretical justification for Faraday’s inferences.12 Plasmonics took a new and comparatively more “high tech” turn in 1957 with Ritchie’s theoretical analysis of energy losses by electron beams passing through thin metal films,13 suggesting that collective surface modes related to but different from the bulk plasmons discussed previously by Pines14 may be excited. Subsequent experimental work by Kretschmann, Raether, and Otto15−17 introduced important optical means of exciting what are now called surface plasmon polaritons or SPPs. With the explosive growth of nanoscience and nanotechnogy in recent times, plasmonics has burgeoned into a vibrant area owing to the many possibilities opened up by both bottom-up and top-down fabrication techniques, as well as modern spectroscopic and imaging methods. Theory and Received: September 28, 2012 Revised: November 30, 2012

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dx.doi.org/10.1021/jp309664c | J. Phys. Chem. C XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry C

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⎛ ω ⎞2 ∇ × ∇ × E(ω) − εr(ω)⎜ ⎟ E(ω) = 0 ⎝c⎠

modeling of plasmonics structures, the topic of the Feature Article, is important for both interpreting experimental results and suggesting optimum configurations, and also for predicting new types of plasmonic behavior for subsequent experimental validation. There is a very significant degree of experimental− theoretical interaction in the field. Many of the examples to be discussed below are in fact joint experimental and theoretical endeavors. Section 2 discusses the basic equations and computational methods relevant to the theory and modeling of plasmonic structures. Simple expectations for plasmonic properties based on limiting cases can be very useful in analyzing numerical results, and some of these “fundamentals” are outlined in section 3. A variety of examples are discussed in section 4. Finally, section 5 presents some brief concluding remarks.

A problem is defined by specifying the spatial and possibly frequency variation of εr or equivalently the refractive index nr = (εr)1/2. For example, referring to Figure 1, εr may take on

2. BASIC EQUATIONS AND COMPUTATIONAL METHODS 2.1. Classical Electrodynamics. Consider an arbitrary material system, possibly containing a number of different components. Electromagnetic waves can be incident upon the system from the far field and be scattered and absorbed by the system, or there could be some source of radiation present such as a dipole emitter. The evolution of the electromagnetic fields is determined by the Maxwell equations, which, for a system containing no naturally magnetic components, take on the form3,18 ε0

∂E(x , t ) ∂P(x , t ) = ∇ × H (x , t ) − ∂t ∂t

μ0

∂H(x , t ) = − ∇ × E(x , t ) ∂t

Figure 1. Various structures that can support or generate SPPs. (a) A semi-infinite metal slab with some superstrate above it. (b) A thin metal film sandwiched between a dielectric substrate and superstrate. If εsub > εsup, light incident from below might be able to excite SPPs on the top side if it has the appropriate angle of incidence. A possible propagation vector direction is indicated along with lines normal to it representing the polarization direction, which must be in the x−z plane in order to excite SPPs. Subwavelength structures on the metal surface, such as particles or roughness (c) and nanoholes or slits (d), provide means for exciting SPPs with light emanating from the superstrate side. In these latter two cases, a variety of angles of incidence may excite SPPs, including the normal incidence. In part d, light incident from the substrate could also excite SPPs.

(1)

where the electric and magnetic fields, E and H, and the polarization density, P, depend on the Cartesian space coordinates, x = (x, y, z), and time, t. In the time domain, P defines the nature of the problem. For example, for many ordinary dielectric media, it often suffices to take P(x, t) = ε0χr(x) E(x, t), where χr(x) is a time-independent susceptibility corresponding to the material present at position x; in the time domain, plasmonic materials require a more sophisticated treatment (vide infra). The energy (ℏω) or frequency (ω) domain version of eq 1 is obtained by writing the real, physical fields E(x, t), H(x, t) and polarization density P(x, t) as Fourier integrals of the form F(x, t) = ∫ dω F(x, ω) exp(−iωt), where F(x, ω) is a complexvalued vector (sometimes called phasor). A single-frequency or time-harmonic solution of eq 1 is then F(x, t) = Re[F(x, ω) exp(−iωt)] with F = E, H, or P.18 In addition, the assumption of a linear polarization response with the electric field, P(x, ω) = ε0χr(x, ω)E(x, ω), and introduction of the relative dielectric constant or permittivity, εr(x, ω) = 1 + χr(x, ω), results in

various values depending on what material occupies a given region of space, and in the case of plasmonic materials it is complex and changes as different ω are considered. Equations 2 and 3 are solved subject to assumptions concerning forms of the asymptotic fields and specification of the electromagnetic boundary conditions.3,18 2.2. Computational Methods. Having established the basic time-domain equations, eq 1, and frequency-domain equations, eqs 2 or 3, several rigorous computational electrodynamics approaches will be briefly discussed, with some emphasis on the finite-difference time-domain (FDTD) approach that is used in most of the examples in section 3. In the FDTD method,19,20 all the fields in eq 1 are represented on discretized grids, low-order numerical derivatives are introduced, and discrete steps in time are taken. The spatial grids associated with each field are not the samethey are displaced from one another by fractions of the grid spacingsand a time leapfrog scheme in E(t) and H(t) is used for the time stepping. This allows for a very simple, secondorder accurate in space and time, scheme to be used that also automatically satisfies the conservation of flux. At least in terms of the time stepping, this is similar in spirit to some of the earlier quantum wave packet propagation schemes.21 There are

−iωεr(ω)ε0 E(ω) = ∇ × H(ω) −iωμ0 H(ω) = −∇ × E(ω)

(3)

(2)

where for simplicity the spatial dependences are omitted. Note that the susceptibility, or equivalently permittivity, has now been allowed to depend on frequency. Re-expressing eq 2 as a single differential equation for the electric field results in the vector wave equation B



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(DDA).30,31 Owing to the vectorial nature of the Maxwell equations and the electromagnetic boundary conditions, FEM approaches turn out to be very involved and many researchers who choose to use this method employ commercial software such as COMSOL, although public domain electrodynamics FEM software can also be found (e.g., ref 32). Public domain software31 is available for the DDA, and it is a very useful and convenient approach for describing finite systems such as one or a few nanoparticles or periodic versions of such systems.

other similarities with quantum wave packet approaches. For example, it is advantageous from a computational standpoint to consider a short pulse initial condition that contains a broad band of frequenciesa single propagation then yields information about the whole band of frequencies. However, there are technical differences as well. In many of the problems of interest, for example, the spatial extent of the incident wave upon a sample is very large. A clever technique known as the total-field scattered-field method19 allows one to introduce such spatially extended waves while keeping the actual computational box to be not much larger than the system. As with quantum wave packet approaches, because of the finite size of a computational box, outgoing waves must be absorbed in strips at the outer edges of the box and an approach, termed the perfectly matched layer approach,19,22 has proven to be particularly effective in the FDTD method. (This approach has an analogue in quantum wave packet absorbing techniques.23) An important technical issue should be noted regarding the FDTD method, or any time-domain method, that is particularly relevant to plasmonics problems. For plasmonic materials, the frequency dependence of εr is significant; e.g., LSP resonances occur as εr passes through certain values as ω is varied, and εr is complex-valued with a negative real part and a positive imaginary part. The polarization density P(t) is then given by a convolution integral involving all past electric field values and a time-dependent susceptibility. The latter in principle could be estimated by Fourier transformation of χr(ω), using interpolation/extrapolation of the generally empirically determined εr(ω) values. This is not a very reliable procedure, and the convolution integral is computationally challenging. One approach, that can also circumvent the need of the convolution integral, is to fit the empirical dielectric constant data to an analytical form. In the case of metals, a sum of a Drude and one or more Lorentzian forms for the dielectric constant is often used with the Drude term describing the intraband, lowfrequency behavior and the Lorentzian terms possibly representing interband contributions.19,24 P(ω) = ε0(εr(ω) − 1)E(ω) is then written as a sum of polarization densities associated with each of the terms. Insertion into the top portion of the frequency-domain relation eq 2 and Fourier transformation back into time (making use of ∂/∂t ↔ −iω type relations) then allows, some extra effort is required to describe plasmonic materials in the time domain relative to the frequency domain and care must be taken to be sure that the resulting fit of the dielectric data is sufficiently accurate if a realistic model is sought. With some scientific programming experience, it is possible to develop one’s own FDTD software because of the detailed presentations, e.g., ref 19, of the approach and its relative simplicity compared to other computational electrodynamics approaches. Three-dimensional problems, however, can be computationally demanding, and parallelization of such software and implementation on multicore computers is advisable.25 There are also some very good, public domain versions of FDTD software including MEEP,26 which is used by many researchers. Finally, commercial FDTD software packages with convenient GUIs are also available, including Lumerical27 that has a number of useful features relevant to plasmonics. There are numerous frequency-domain approaches to solving the Maxwell equations. These include the finite-element method (FEM)28,29 and the discrete dipole approximation

3. SURFACE PLASMON FUNDAMENTALS Formulas for the electric fields of surface plasmons and the conditions to excite them are available for various idealized situations. Such equations are invaluable both for developing an intuitive feeling for a given, more complex problem at hand and for assessing the need or correctness of more computationally oriented approaches such as those outlined in section 2 above. 3.1. Surface Plasmon Polaritons (SPPs). Figure 1 displays various simple structures that can support SPPs. Consider a semi-infinite metal slab characterized by complex, frequency-dependent dielectric constant εr = εmet(ω) for z < 0 with some superstrate material of dielectric constant εsup for z > 0, Figure 1a. A time-harmonic solution of the Maxwell equations, eq 1, is given by Re[E(ω) exp(−iωt)], where1,2 ⎛ k ⎞ E(ω) = A⎜x̂ − x z⎟̂ exp[i(kzz + kxx)] kz ⎠ ⎝

(4)

with kx(ω) ≡ k SPP(ω) =

ω c

εsupεmet(ω) εsup + εmet(ω)

(5)

and 1/2 ω ⎛⎜ εsup(z > 0) or εmet(ω)(z < 0) ⎞⎟ kz(ω) = ⎜ ⎟ c⎝ εsup + εmet(ω) ⎠

(6)

The time-average of the square of this solution, proportionate to the intensity, is |E|2. SPPs arise from the peculiar optical properties of metals in certain spectral regimes wherein εmet has a negative real part and positive imaginary part, making eqs 5 and 6 complex-valued. The equations and conditions for SPPs are most easily understood in the limit when Im[εmet] is small enough relative to Re[εmet] to be neglected, εsup is a real, positive number, and Re[εmet] < −εsup. It is then the case that kSPP is real and positive and kz is imaginary. This represents an SPP propagating with propagation vector kSPP along the x direction and exponentially decaying (evanescent) along z as |z| increases. Equation 4 shows that the SPP has a longitudinal component, i.e., a nonzero field component (Ex) in the direction of propagation, which distinguishes it from ordinary light waves. However, there is still a significant transverse component, Ez, proportionate to kx/kz, which is sometimes not appreciated. It is also not hard to see that the SPP propagation vector kSPP > (ω/ c)(εsup)1/2. The right-hand side of this inequality is the wave vector magnitude for ordinary light waves propagating in the superstrate medium, and the fact that the SPP propagation vector is larger than the ordinary light wave propagation vector implies that it can neither emit ordinary light waves into the superstrate nor can it be excited by ordinary light waves emanating from the superstrate side, since the appropriate wave C



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Figure 2. Examples of Re(E) and |E|2 obtained from eq 1 assuming gold for z < 0 characterized by the complex dielectric constant inferred from ref 34 and air (εsup = 1) for z > 0 (with A taken to be 1 arbitrary field unit). The red dashed curve indicates the z = 0 metal/air interface. Upper panels a−c correspond to Re(E(t=0)), and lower panels d−f correspond to |E|2. In parts a−c, the black curves are nodal lines. In parts d−f, the black contour lines represent when |E|2 is at 1/e of its maximum. Parts a and c correspond to λ = 532 nm (=2.33 eV); parts b and e correspond to λ = 633 nm (=1.96 eV); parts c and f correspond to a (0, 1) BW-SPP at λ = 628 nm consistent with a periodic structure with P = 600 nm.

the SPP propagation direction and the normal to the surface the SPP is propagating on, e.g., the x−z plane in Figure 1. Periodic arrays of objects patterned on a metal film, e.g., periodic arrays of holes (periodic analogue of Figure 1d) in a metal film, are especially useful for exciting SPPs, since largearea SPP excitations can be created with relative ease. The periodicity makes the structure a grating and gratings lead to diffracted light, including light diffracted along the surface of the metal film structure, with altered wave vectors relative to the incident light. For simplicity, consider the case of a square array of holes in a metal film that has the same periodicity P along both the x and y directions. The diffracted wave vector parallel (∥) to the grating or metal surface is2,33

vector components cannot match. The SPP could therefore be termed a “dark mode” in that it would not be visible to the eyes of an observer in the far field from the superstrate side without some additional features present in the system to couple the SPP into light. There are many ways of exciting SPPs, all of which require going beyond the simple setup in Figure 1a. The essence of all these approaches is wave vector matching, which is also termed phase or momentum matching, such that there are incident electromagnetic waves, or generated electromagnetic waves within the system, that have wave vectors that match the SPP propagation vector, kSPP (or its real part if it is complex). One way is to consider a thin metal film with a dielectric medium substrate below it that has dielectric constant εsub > εsup, Figure 1b. It can then be possible to match the x-component of the incident light wave vector with kSPP by tilting the light at some angle θSPP relative to the surface normal such that ((εsub)1/2ω/ c) sin(θSPP) = kSPP. With eq 5, this expression may be rearranged to read sin(θSPP) =

k diff = k inc +

(8)

where mx and my are integers. If one focuses on normally incident light that has no parallel incident component, then kdiff =

Re(εmet)εsup εsub(Re(εmet) + εsup)

2π (mx x̂ + my ŷ) P

2π (mx x̂ + my ŷ) P

(9)

Equating the magnitude of eq 9 with the SPP propagation constant kSPP then leads to a useful equation for predicting the incident wavelength such that SPPs may be excited33

(7)

where I have again assumed that Im(εmet) is much smaller in magnitude than Re(εmet). This structure for exciting SPPs is also called the Kretschmann configuration or attenuated total reflection (ATR).15,16 The term ATR arises because the actual angle (which may deviate slightly from the predicted value, eq 7) will occur when the reflected light is severely reduced (attenuated), indicating the energy is going into SPP generation. (A related configuration for exciting SPPs is the Otto configuration.17) It is also possible to excite SPPs from the superstrate side by making use of subwavelength structures such as nanoparticles, Figure 1c, or holes or slits, Figure 1d, or simply surface roughness.1,2 In these cases, scattering events can lead to the generation of higher wave vector components. The polarization of the exciting electromagnetic waves is also important: they must have polarization in the plane defined by

λ=

2πc = ω

P 2

mx + my

2

εX Re[εmet(λ)] εX + Re[εmet(λ)]

(10)

In eq 10, X could be either sup or sub, since SPPs on either the top of the metal film, interfacing with the superstrate, or SPPs on the bottom of the metal film, interfacing with the substrate, could conceivably be excited. Equation 10 is an implicit equation: Given a particular metallic dielectric constant function, specification of whether or not the superstrate or substrate SPPs are being sought, and particular integer values of mx and my, one must find the value of ω, or equivalently λ, such that eq 10 is true. This can be accomplished by rewriting eq 10 in the form f(λ) = 0 and treating it as a root-finding problem. D



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is being used to generate the SPPs. Referring to Figure 2b, for example, the SPP intensity in the superstrate ∝ εsupA2 that results can then be larger than that for the incident light in the substrate, εsubA02, and the ratio, g2 = εsupA2/(εsubA02), is called the intensity enhancement factor. Raether2 has discussed how g2 depends on the thickness of the metal film and also the incident light wavelength, which determines εmet. For visible light, 10−100 nm thick silver and gold metal films typically exhibit g2 values in the 10−100 range. These values of g2 are lower than those achievable in localized surface plasmon excitations of clusters of metal nanoparticles. 3.2. Localized Surface Plasmons (LSPs). The problem of plane waves interacting with a uniform sphere of arbitrary complex relative dielectric constant εmet, surrounded by a uniform medium of relative dielectric constant εmed, was analytically solved over 100 years ago by Mie.12 A number of other relevant problems, such as multilayer spheres3 and spheres exposed to radiation from an emitting dipole35,36 as opposed to plane waves, also admit analytical solutions. “Mie theory”, as such solutions are collectively called, remains an extremely valuable tool for many research-level problems in plasmonics. However, its formal expressions, involving series of complex Bessel functions, are not very instructive and the simpler, quasistatic limitting form1,37 valid for small particle sizes compared to incident wavelength is much more useful from a pedagogical standpoint. Consider incident plane wave light, with fixed wavelength λ, on a metallic sphere emersed in a uniform medium. If λ = 2πc/ ω ≫ a, where a is the radius of the sphere, then the electric field outside the sphere reduces to a simple result that corresponds to the sum of the incident field E0 plus that due to the dipole p induced in the sphere. Keeping the convention that the time-harmonic solution of eq 1 is Re[E(ω) exp(−iωt)], this corresponds to

Usually the structure of the dielectric constant is such that only one root exists that is in the spectral region of interest. Thus, while purely a classical electromagnetic phenomenon, one has the result that the SPPs in periodically structured metal films are predicted to exist at discrete wavelengths corresponding to the integer pairs (mx, my). Of course, in practice, coupling between top and bottom SPPs, as well as the presence and coupling to other features such as localized surface plasmons not related to the periodicity, leads to a broadening of the discrete lines predicted by eq 10. SPPs that can be generated in periodic structures via the diffraction discussed above are termed Bloch wave SPPs or BWSPPs. The electric fields associated with BW-SPPs, in the case of normal incidence, are given by superpositions of the propagating SPPs, eq 4, that become standing waves. For example, consider normally incident light propagating along z and polarized along x interacting with the periodic analogue in Figure 1d. Suppose this incident wave has a wavelength that satisfies eq 10 for the case mx = ±1 and my = 0. It is then the case that SPPs can be excited with kx = mx 2π/P = +2π/P and −2π/P equally with the result that these counter-propagating waves (using eq 4) form a standing wave x-dependence proportionate to cos(kxx). We conclude this introduction to SPP basics with several examples of SPPs along a gold/air interface, Figure 2. The upper panels, Figure 2a−c, are instantaneous snapshots of the actual electric field at t = 0, Re[E(x, ω)], and the lower panels, Figure 2d−f, are the corresponding electric field intensities, |E|2. Panels a and d correspond to λ = 532 nm, and panels b and e correspond to λ = 633 nm. (λ = 532 and 633 nm are commonly used green and far red laser lines, respectively.) These images correspond to ordinary propagating SPPs generated by eq 4 using the dielectric constant data for gold of Johnson and Christy.34 The SPP propagation length, L, associated with the propagation distance such that |E|2 has been reduced by 1/e from its maximum value is L = 0.5/Im(kSPP). Inspection of the lower panels in Figure 2 shows both L and the extent of the near-field intensity along z for SPPs at λ = 633 nm are significantly better than those at λ = 532 nm. This is because | Re(εAu)/Im(εAu)| ≈ 9.6 at 632 nm, compared to a value of ∼2.1 at 532 nm; i.e., the real part of the metal’s dielectric constant is significantly larger in magnitude than the imaginary part at the higher wavelength. This is generally true, since as one considers ever higher wavelengths, or lower photon energies, the dielectric constant of a metal approaches the Drude limit and its real part approaches negative infinity while its imaginary part remains bounded. Notice also that the nodal lines in panel a are more slanted than those in panel b. This too reflects the relative poor quality of the SPPs at 532 nm compared to those at 633 nm and is a manifestation that Im(kSPP) cannot be neglected in this case. Panels c and f correspond to a BW-SPP with mx = ±1 and my = 0. It turns out that the solution of eq 9 yielded λ = 628 nm, a value accidentally close to the propagating case in panels b and e. It is interesting to note that the intensity pattern of the BW-SPP, panel f, is different from the propagating case closest to it, panel e. In the case of propagating SPPs, this latter quantity is proportionate to cos2(kxx − ωt), which timeaverages to a constant (1/2), whereas for the BW-SPP it is proportionate to cos2(kxx) cos2(ωt), which time-averages to cos2(kxx)/2. The various images in Figure 2 were all constructed assuming maximum field amplitude A = 1 arbitrary field unit. It is often the case that A is larger than that for the incident light, A0, that

E(ω) = E0(ω) +

3(p(ω) ·r)̂ r ̂ − p(ω) 1 4πε0εmed r3

(11)

with the dipole p(ω) = εmedα(ω)E0(ω)

(12)

and polarizability ⎛ ε (ω) − εmed ⎞ α(ω) = 4πε0a3⎜ met ⎟ ⎝ εmet(ω) + 2εmed ⎠

(13)

(To recover the traditional result of the Gausian units system in which α has units of volume, one sets 4πε0 = 1.) For the specific case of the incident light being z-polarized and propagating along the x axis, eq 11 then becomes ⎛ 3z(x x̂ + y ŷ + z z)̂ − r 2 z ̂ ⎞ ⎟ E(ω) = E0⎜z ̂ + α(ω) r5 ⎠ ⎝ × exp(ik medz)

(14)

with kmed = (εmed)1/2ω/c. The quasistatic-limit extinction spectrum, which for very small particles is the same as the absorption spectrum, is given by σext = (kmed/ε0)Im(α), or σext(ω) = 4πa3 εmed E

ω ⎛ εmet(ω) − εmed ⎞ Im⎜ ⎟ c ⎝ εmet(ω) + 2εmed ⎠

(15)



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The LSP is a resonance that is considered to be fully excited at a frequency such that eq 15 is a maximum. Equation 14 shows that the field strength will also be a maximum at this frequency. For the case Im(εmet) small compared to |Re(εmet)|, the condition for the LSP to be fully excited is thus Re(εmet) = −2εmed. In contrast, the SPP dispersion relation, eq 5, shows that SPPs can in principle exist over a continuous range of frequencies, although depending on the excitation mechanism one may resonantly tune into an SPP at a particular frequency by varying some parameter such as an angle of incidence or grating period. Figure 3 is an image of electric field intensity for an a = 15 nm gold sphere in water with incident vacuum wavelength λ =

Figure 4. Light incident from a glass substrate below a periodic array of nanoholes in a 100 nm thick gold metal film (lower schematic) leads to a structured transmitted light spectrum. The calculated FDTD transmission (circles) is very well fit by a multiple Fano resonance model (solid curve). See text for further details. The transmission spectrum is reproduced with permission from ref 41.

Figure 3. Near-field electric-field intensity of a 30 nm diameter gold sphere in water. The black contour line corresponds to when the intensity is 1/e of its maximum value.

535 nm, which is close to the maximum in the absorption spectrum. One can see the intense dipolar near-field intensity of the surface plasmon resonance. In this case, the maximum intensity enhancement is not especially largeabout a factor of 40. Other spherical structures, however, such as dielectric-core and metal-shell structures tuned to have resonances in the infrared region where the plasmon quality factor is particularly high, can have considerably larger intensity enhancements.7 Also, nonspherical structures with large aspect ratios or sharp features, e.g., bipyramidal nanoparticles,38 can also have considerably larger intensity enhancments in the 100s. Dimers and clusters of metal nanoparticles in general can have considerably higher enhancements, e.g., refs 39 and 40.

Wfilm(λ)]/(Iincπa2), where Wfilm is the power that results from a calculation corresponding to the metal film with no holes present, Iinc is the incident intensity, and Iincπa2 is the power incident underneath each hole. Thus, Wnorm ≈ 1 represents near perfect transmission of the incident power underneath the holes. The fact that Wnorm can be a significant fraction of unity and even slightly exceed it (due to wave interactions absent from its geometric-optics-based definition) is remarkable in view of the fact that, for the entire wavelength range, λ > 2a. Within a geometric optics picture, there should be no transmission at all. The phenomenon evident in Figure 4 was first observed in 1998 by Ebbesen and co-workers,33,42 who termed it “extraordinary optical transmission” or EOT. Since then, there has been much discussion about the mechanism underlying EOT with the review by Garcia-Vidal et al. being an excellent overview.43 One can achieve EOT via coupling into electromagnetic modes or resonances that trap light energy in the structure. In particular, if SPPs can be excited, they can be responsible for EOT. It turns out that the strong spectral features in Figure 4 can be correlated with the wavelengths predicted by eq 10 corresponding to various BW-SPP excitations.41 These features, particularly when well separated, are seen to be asymmetric structures composed of a minimum (that is almost zero) followed by a maximum, and then a lingering tail. The resonance structure with a minimum/ maximum pair of 676 nm/710 nm is a good example. These types of resonance features can be well described by the Fano line shape form that originally arose in the context of atomic physics.44 The Fano line shape arises from the interaction of a

4. ILLUSTRATIVE EXAMPLES 4.1. Surface Plasmon Polaritons in Metal Film Structures. Consider a 100 nm thick gold film on top of a glass substrate, with a periodic array of holes of radius a = 100 nm with air above as schematically illustrated in the lower portion of Figure 4. With light normally incident from the glass side, as indicated, FDTD calculations were carried out to infer the spectrum of transmitted light indicated in the top portion of Figure 4.41 In the calculation, a short pulse containing a broad range of frequencies (or wavelengths) interacts with the metallic structure and the time-evolving electric and magnetic fields are Fourier transformed to yield frequency-dependent fields. The Poynting vector or flux is obtained from these electric and magnetic fields, and its component normal and pointing up into the air is integrated across the area above the structure to yield the outgoing power, W. The normalized transmission in Figure 4 is defined to be Wnorm = [W(λ) − F



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chemical imaging.4 Of particular interest here are the combined experimental and theoretical studies of plasmonic crystals,47−54 metal films deposited over polymer substrates with patterned nanowells or nanoposts in them. The resulting nanowell or nanopost arrays, depending on the deposition technique, can offer more structural features than the simpler nanohole arrays discussed so far. For example, the walls and bottom of a nanowell can also be coated with metal film. In this work, a simple wavelength-integrated transmission was found to be a good figure of merit for refractive index sensing and both BWSPP and LSPs were found to be important contributors to the system response. There are many system parameters to consider in such systems: periodicity, nanohole or nanopost diameters, depths or heights, thicknesses of various layers, etc. The FDTD simulations can be computationally very demanding, so systematically considering all possibilities is generally not feasible. Surrogate-based optimization procedures represent one route to such multidimensional optimizations.50,55,56 These procedures involve constructing a computationally fast surrogate or model of the system’s response function and using the surrogate to predict the parameters that optimize a desired response. Subsequent full simulation calculations are used to improve upon the surrogate and the estimate of the optimum set of system parameters. Initially, relatively simple quadratic surrogate models were employed,50 and more recently, more sophisticated Gaussian process models55,56 have been explored. There are a variety of other optimization procedures that might also be very useful for such purposes including, for example, genetic algorithms, e.g., refs 57 and 58. To conclude this section on plasmonic structures that support SPPs, I would like to discuss a theoretical proposal for remotely exciting SPPs with a structure called the remote grating or r-grating,59 Figure 6a. In the r-grating structure, one

zero-order bound or metastable state with a zero-order continuum or at least much broader zero-order metastable state. In the present case, the narrow zero-order state is a BWSPP and the zero-order continuum of states corresponds to light that directly penetrates through the metal film. The solid curve in Figure 4 is the result of fitting a multiple Fano resonance model to the FDTD transmission data given by circles. To further illustrate the nature of the BW-SPPs, Figure 5a−f shows electric field intensities inferred from the FDTD

Figure 5. Panels a, b, ..., f correspond to electric field intensities at λ = 610, 620, ..., 660 nm for light interacting with a periodic nanohole array of Figure 4. Reproduced with permission from ref 41.

calculations at λ = 610 nm, 620 nm, ..., 660 nm. Equation 9 suggests that a zero-order BW-SPP would occur on the air side of the metal film at 629 nm corresponding to mx = ±1, my = 0 and the actual transmission shows a minimum/maximum feature at 628 nm/640 nm. One can see in Figure 5 this resonance being excited, with the highest intensity near 640 nm, Figure 5d. Equation 9 also predicts that there will be a zero-order glass-side BW-SPP at 688 nm corresponding to mx = ±1, my = ±1, and Figure 5e and f, corresponding to λ = 650 and 660 nm, shows the beginning of that excitation. BW-SPPs are not always responsible for spectral features in transmission spectra. LSPs associated with individual holes can also play an important role in shaping the overall spectrum,41 and other features, including diffraction, can be important. In particular, Raleigh anomalies (RAs), sometimes called Wood anomalies, although the latter represent a broader class of diffractive resonances, can sometimes be evident. The RA corresponds to the threshold of a diffraction order and represents light (not SPPs) being diffracted at an angle parallel to the metal surface. The condition for an RA is simply λRA = PnX/(wx2 + wy2)1/2, where wx and wy are integers, nX is the refractive index of the substrate or superstrate, and P is the periodicity of the nanohole structure. This condition is similar in spirit to that for BW-SPPs, eq 9, and often leads to RA positions very close to BW-SPPs. Often, the BW-SPPs tend to be stronger and broader in character and mask any RA features. Nonetheless, refs 45 and 46 showed, both theoretically and experimentally, how one could gradually alter the superstrate refractive index and cause an RA associated with one side of the structure to “pass through” a BW-SPP on the opposite side. When overlapping, the resonance was dubbed an RA-SPP. As the RA-SPP is approached by variation of a refractive index, there can be a rapid rise in transmission over a narrow range of wavelengths, which is relevant to refractive index (RI) sensing. There has been a lot of work concerning hole arrays and related structures in metal films of relevance to sensing and

Figure 6. The r-grating structure for excitation of SPPs is outlined schematically in part a, and the resulting electric field intensity map inferred from an FDTD calculation corresponding to incident light with λ = 811 nm (which leads to the strongest response) is shown in part b. Part b is reproduced with permission from ref 59.

has a dielectric slab substrate (e.g., glass or some polymer) that could be relatively thick and a thin plasmonic metal film on top of it. Air or some other superstrate with relative dielectric constant less than that of the dielectric slab is above the metal film. A grating is embedded in the dielectric slab, and here it is taken to be simply a periodic array of metal bars separated by periodicity P, as indicated in Figure 6a. With light normally incident upon the grating from below, light will be diffracted up into the dielectric slab with angle θGM given by G


The Journal of Physical Chemistry C sin(θGM) =

mλ Pnsub

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from the emitter, if it is placed close enough to the nanoparticle, provides the higher magnitude wave vector components required to excite a dark mode. Figure 7 displays results of FDTD calculations obtained for a dipole emitter close to a gold bipyramidal nanoparticle.65 The

(16)

where λ is the vacuum wavelength of the light, nsub is the refractive index of the dielectric slab, and m is an integer. (Note the Rayleigh anomaly discussed above in the context of the RASPP corresponds to θGM = π/2, although the RA-SPP is not of relevance to the present discussion.) The idea is to simply engineer the periodicity, or perhaps refractive index, such that a grating mode angle determined by eq 16 equals the corresponding angle required to excite SPPs on the top of the flat metal surface, eq 7. Equating eqs 16 and 7, in fact, leads to none other than eq 10, the equation for the generation of BW-SPPs with X = sup and mx = m, my = 0. (The grating in the r-grating structure of Figure 6a is invariant with respect to y; i.e., the metal bars are infinitely extended out of the plane. A more complex 2-D grating structure could be used such that the more general form of eq 10 would result.) Thus, we have a procedure for exciting BW-SPPs with the grating structure not on the metal surface as it is in the nanostructured metal films we have been discussing. Another angle (so to speak) is provided by h. Fabry−Pérot (FP) waveguide modes within the substrate can also be excited when h is chosen to be consistent with the associated FP mode angle, θFP.59 Having all these resonances occur simultaneously could lead to significant enhancement in SPP field strengths and higher quality (narrower) spectral response. Furthermore, having SPPs generated on a flat film, as opposed to nanostructured ones, will reduce scattering losses. Figure 6b illustrates a result achieved using FDTD simulations with εsup = 1 (air), εsub = 2.31 (glass), and a 40 nm thick gold metal film. The grating was also taken to be gold bars with periodicity P = 800 nm, and h was slightly over a micrometer, h = 1050 nm. By scanning the incident wavelength, the intensity measured 10 nm above the metal film was found to display a narrow-band response (fwhm = 3 nm) with peak intensity enhancement of almost 250. Calculations corresponding to a traditional ATR configuration led to a relatively broad-band response (fwhm = 100 nm) and a peak intensity enhancement of 245. Due to the improved SPP quality and narrow-band response, the r-grating could also find uses in more complex, hybrid structures (see next subsection) . 4.2. Localized Surface Plasmons and Hybrid Structures. Ordinary light is not the only means of exciting SPPs or LSPs. Fluorescing molecules, for example, near metal nanostructures can couple into plasmons.60,61 While a full quantum mechanical treatment is, of course, the most ideal approach, some insights can be gained with simpler classical electrodynamics models. One simple way of modeling a fluorescing molecule or perhaps quantum dot in electrodynamics is to treat it as a radiating point dipole. Analytical solutions for the case of a point dipole and a sphere35,36 have helped in the analysis of experimental measurements of photoluminescence enhancements and quenching in quantum dot−gold nanoparticle systems.62,63 It is straightforward to incorporate a radiating dipole into a simulation approach such as the FDTD method, so more challenging cases, e.g., dimers64 or nonspherical particles,65 can be studied. An interesting feature of this way of coupling into plasmons is that “dark plasmons”, i.e., modes that exist but do not couple to ordinary light, may be excited. This situation is analogous to ordinary SPPs on flat surfaces not directly coupling to ordinary light and therefore requiring special wave vector matching means of being excited (section 3.1). In the present case, the emission

Figure 7. Panel a displays the absorption spectrum for a bipyramidal gold nanoparticle (black dashed curve) when exposed to ordinary light polarized along the nanoparticle’s long axis, and the relative radiative (solid black curve) and nonradiative (red dashed curve) when, instead of ordinary light, a dipole emitter, also polarized along the long axis, is close to the tip of the nanoparticle. See the schematic diagram in the upper right corner. Panels b and c are the real parts of the frequencyresolved electric fields at the dipolar and dark quadrupolar energies, respectively. Reproduced with permission from ref 65.

nanoparticle is 83.4 nm long and has a maximum width of 30 nm, and in all cases, the radiation is polarized along the particle’s long axis. The dashed black curve in Figure 7a is the absorption spectrum that results from ordinary radiation (not the emitter) interacting with the system. It displays one prominent peak near 1.5 eV (826 nm) that is associated with the usual, dipolar LSP excitation. However, if one replaces the plane wave illumination, consistent with ordinary optical excitation, with a dipole emitter 13 nm from the tip of the particle, as indicated in Figure 7a, and uses FDTD to calculate the relative radiative (Γr/Γ0) and nonradiative decay rates (Γnr/ Γ0) of the system, one obtains the solid black and dashed black curves. These decay rates are relative to the radiative decay rate (Γ0) of the isolated dipole emitter and are obtained by Fourierresolving the FDTD fields and carrying out appropriate surface integrals involving the Poynting vector.65 Γnr/Γ0 is of special interest because it reflects energy being absorbed by the nanoparticle. In particular, Γnr/Γ0 shows a prominent secondary peak near 1.75 eV (708 nm) that is absent in the absorption spectrum. This peak is identified with a dark, quadrupolar LSP H



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mode. Plots of Ez, the component of the electric field along the long axis of the bipyramid, Figure 7b and c, respectively, are consistent with the assignments of dipolar and quadrupolar modes.65 Reference 65 went on to show that by lining up several bipyramidal nanoparticles along a line that it is possible to enhance the propagation length along the structure by having the dipole emitter excite at a quadrupolar resonance energy as compared to the dipolar resonance energy. The bipyramidal gold nanoparticles discussed above may seem esoteric. However, there are straightforward experimental procedures for generating monodisperse colloidal solutions of these nanoparticles.66 As noted earlier, their relatively sharp tips are promising for field enhancements and also for manipulation with optical force techniques. The FDTD method has been used to calculate the optical force on one bipyramid due to the presence of another bipyramid when these dimers are under illumination67 and to determine the forces acting on bipyramids when they are in aqueous solution but close to a glass/water interface with total internal reflection (TIR) illumination.68 Depending on the details, the forces can be either negative (attractive) or positive (repulsive). The TIR calculations, which were part of a joint experimental−theoretical effort, were used to interpret the remarkable experimental observation of the selective deposition on the glass of the bipyramids compared to spherical particles also present in the colloidal solution. The absorption, as opposed to emission, of an object like a quantum dot (QD) placed near a metal nanoparticle structure also provides for some interesting effects. The top of Figure 8

quantum dot is introduced, however, the result is found to be the solid black squares. Quite remarkably, a strong dip or “transparency” is seen to occur. The effect is strongest when the isolated quantum dot absorption is much narrower than that of the LSP. In the example, the QD decay rate is hγQD = 0.002 eV, which can be compared to the LSP full width at half-maximum, which is about 0.1 eV. The solid red curve is the result of a simple analytical model corresponding to two coupled dipoles,69 somewhat akin to the Gersten−Nitzan model70 for a molecule interacting with a plasmonic particle. The dip arises from destructive interference between the QD and dimer plasmon, and it can also be thought of as a variation on the Fano resonance effect discussed earlier in the context of nanohole arrays. As such, the phenomenon or related phenomena occur in other systems.71−73 Interestingly, a paper treating similar problems but using a multipole expansion approach to solving the Maxwell equations appeared almost at the same time as our own effort, and it also came to similar conclusions.74 It should be noted that there has been some earlier work, based on approximate quantum mechanical models, quantum dot, or exciton coupling with plasmonic systems, that has noted the possibilities of asymmetric line shapes and transparency dips.75,76 More recently, a quantum mechanical Green function approach, capable of describing strong field interactions, was also introduced and used to study these types of problems.77 In the above QD structures, the QD was an active participant in the system response, changing it dramatically by allowing dark modes to be seen and causing transparencies. It is also possible to incorporate absorbing species into a plasmonic system with a more benign, but still very useful, effect. In particular, a metal nanostructure could be coated with a thin layer of a polymer containing a photosensitive dye such as an azobenzene derivative. After exposure to light, the resulting surface topography is a measure (roughly a negative image) of the near-field intensity patterms. Comparison of experimentally deduced topographies with the result of FDTD calculations validates this picture.78 A recent application of this approach to gold nanocubes identified not only electromagnetic hot spots but spatially confined cold spots.79 Finally, on this subject of imaging near fields with photosensitive materials, it should be noted that multiscale theoretical simulations have also been developed that can predict the nuances of the topological changes to the photosensitive polymer layer.80,81 Another type of hybrid system that naturally suggests itself is to combine a grating structure that could lead to BW-SPPs with metal nanoparticles that have LSPs. Engineering a system such that the corresponding resonances coincide, and perhaps even introducing other resonant features, is another avenue to pursue that theory and modeling can contribute to by suggesting optimum configurations. Promising theoretical results have been obtained with nanoparticle-filled grating structures,82 and recent combined experimental and theoretical work83 on hexagonal gold nanoparticle arrays leading to highfidelity SERS response are steps in this direction. Many of the examples above have been relatively sophisticated plasmonic structures. However, to come full circle, the most basic of plasmonic systems, colloidal gold (see section 1), still can yield surprises. Experimental results (symbols) for the maximum absorption wavelength, λmax, for various diameter, d, gold nanoparticles in various solvents are displayed in Figure 9A. The remarkable feature of these results is that there is a strong red shift in λmax as d is decreased below

Figure 8. Illustration of a quantum dot induced transparency for the silver nanoparticle dimer/quantum dot system schematically indicated. Panel a is the extinction spectrum, and panel b is the scattering spectrum. Filled, black square circles are FDTD results for the full structure, and filled, green circles are FDTD results for only the silver dimer. The red curves are the result of a simple coupled dipole model. Spectra are reproduced with permission from ref 69.

sketches a structure composed of two silver ellipsoidal nanoparticles with an absorbing QD such as a CdSe nanocrystal between them.69 The calculations involved silver spheroids of length 40 nm and diameter 8 nm and a spherical QD of radius 4 nm. The region of space occupied by the QD is taken to be a Lorentzian medium that has parameters consistent with cryogenic conditions where there would be a narrow line width. The QD is assumed to have been size-selected such that its absorption is very close to that of the LSP associated with the metal nanoparticle dimer. See ref 66 for details. Parts a and b of Figure 8 depict extinction and scattering spectra, respectively, obtained from FDTD calculations. The solid green circles correspond to the extinction spectrum for the system not allowing the quantum dot to absorb. It corresponds to a single peak reflective of the strong, dipolar LSP resonance that is present just above 2.1 eV (590 nm). When the absorbing I



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The emergence of plasmon resonances themselves from quantum mechanics is also a fascinating avenue to explore.97 For example, based on time-dependent configuration interaction calculations, electron density oscillations consistent with a simple surface plasmon model were identified in the electronic dynamics of linear hydrogen chains after exposure to a short, intense laser pulse.98

5. CONCLUDING REMARKS The theory and modeling of plasmonic structures has advanced considerably over the past few years. Rigorous computational electrodynamics simulations are now routinely carried out in conjunction with experimental work, and this Feature Article included a number of such examples. Purely theoretical predictions for plasmon generating structures such as the rgrating and for several remarkable optical phenomena due to coupling with quantum dots were also noted. The hybrid quantum dot plasmon systems, and the general push to smaller nanostructure features and components, mean that the rigorous incorporation of quantum mechanics in plasmonics will become an ever important direction to pursue.

Figure 9. Panel A displays experimental absorption maximum wavelengths as a function of gold nanoparticle diameter (symbols) for colloidal gold in the indicated solvents. Solid curves are the result of a simple model. Panel B is a schematic of a small gold nanoparticle (gold-orange) surrounded by a layer of surfactants. Reproduced with permission from ref 84.

10 nm. One would have expected, based on Mie theory, that as d decreases λmax would decrease (blue shift) and then plateau to the constant, small-particle limiting value predicted by eq 15. A variety of possibilities for the origins of the red shift or reversal of the surface plasmon resonance as ever smaller particles are considered were pursued.84,85 A very simple model, based on the schematic in Figure 9B, was ultimately found to account for the observations in Figure 9A and also was used to predict the outcome of additional experiments.84 The surfactant molecules surrounding the nanoparticle interact with the surface gold atoms, and these interactions become ever more important as the surface to volume ratio increases with ̈ level, these interactions “tie up” decreasing d. At a naive some conduction electrons in the metal, and effectively the conductivity of a thin (1 nm) outer layer of gold atoms may be lowered by a small amount. Once this ansatz is made, it is relatively straightforward to develop an analytical model consistent with this that utilizes multilayer Mie theory. The results of this model are the solid curves in Figure 9. The correctness of the model proposed to account for the anomalous blueshift in λmax, and the role of other quantum mechanical effects, can probably only be ascertained by detailed quantum mechanical calculations and further experiments yet to be carried out. In fact, a frontier in the theory and modeling of plasmonic nanostructures is the incorporation of quantum mechanical phenomena. Another phenomenon due to quantum mechanics is the possibility of a nonlocal optical response in the metal, which essentially corresponds to the polarization at a given spatial point depending not only on the local electric field but on the electric field in its neighborhood. One phenomenological way of accounting for nonlocal response is with the hydrodynamic Drude model. 86 A means of incorporating the hydrodynamic Drude model within FDTD has been introduced,87 and several studies have shown that there can be important nonlocal effects for small metal nanoparticles, introducing volume plasmon resonances and reducing the electromagnetic field enhancements in very small gaps between particles.88−90 It should be noted, though, that this work implicitly employed a curl-free approximation and it has been shown that relaxing this approximation leads to fewer volume resonances.91,92 Very interesting recent theoretical93,94 and joint experimental−theoretical95,96 work showing quantum limitations to plasmonic enhancements and quantum size effects should also be noted.

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AUTHOR INFORMATION

Notes

The authors declare no competing financial interest. Biography

Stephen K. Gray obtained a BSc. in Chemistry from Carleton University, Ottawa, in 1977, carrying out honours thesis work with Prof. James S. Wright. In 1982, he obtained a Ph.D. in Chemistry from the University of California, Berkeley, under the supervision of Prof. William H. Miller. He subsequently carried out postdoctoral studies with Profs. Mark S. Child at Oxford University and Stuart A. Rice at The University of Chicago. He was an Assistant Professor of Chemistry at Northern Illinois University from 1986 to 1990 before joining the scientific staff at Argonne National Laboratory. Dr. Gray’s research interests have ranged from classical, semiclassical, and quantum theories of chemical reaction dynamics to optical interactions in nanostructures. He is currently a Senior Scientist and Group Leader of the Theory and Modeling Group at Argonne’s Center for Nanoscale Materials.

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ACKNOWLEDGMENTS I have been very lucky in having an army of theoretical and experimental collaborators who have aided and abetted this work. On the theory side, I am in debt to S.-H. Chang, M. J. Davis, A. E. DePrince III, T.-W. Lee, J. M. McMahon, R. L. Miller, J. M. Montgomery, and G. C. Schatz. On the J



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experimental side (although many here have contributed essential theoretical ideas and calculations as well), I am in debt to R. Bachelot, A. Chen, M. Chowdhury, M. L. Juan, J. R. Lakowicz, X. Q. Li, M. Z. Liu, J. Maria, R. A Nome, P. GuyotSionnest, R. G. Nuzzo, T. Odom, M. Pelton, S. Peng, J. Plain, Y. Sun, D. Ratchford, J. A. Rogers, N. F. Scherer, V. Vlasko-Vasov, U. Welp, G. P. Wiederrecht, and X. Wu. I thank G. P. Wiederrecht for reading and commenting on the manuscript. This work was performed at the Center for Nanoscale Materials, a U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences User Facility under Contract No. DE-AC02-06CH11357.

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