Band Structures of Two Dimensional Surface Plasmonic Crystals Solid States Physics
Jimmy Zhan Dr. Marc Dignam November 2013
Abstract The dispersion relations and radiation properties of two-dimensional surface plasmonic crystals were studied via the Rigorous Coupled Waved Analysis (Fourier Modal Method). In particular, properties of hexagonal (triangular) lattices with cylindrical protruding pillars (called positive plasmonic crystals) and recessed air holes (negative plasmonic crystals) were investigated for their dispersion relations, mode degeneracies, radiation properties, and plasmonic band gaps. It was found that the optical properties of the crystal are highly dependent on the size and shape of these cylindrical pillars (or holes). Near the Ξ point, there are a total of three radiative modes for both positive and negative plasmonic crystals. Two of these three modes couple to both p and s polarized light, whereas one of the modes only couple to p polarized light.
Table of Contents Abstract ......................................................................................................................................................... 1 Introduction .................................................................................................................................................. 3 Theory ........................................................................................................................................................... 5 Rigorous Coupled Wave Analysis (Fourier Modal Method)...................................................................... 5 Results ........................................................................................................................................................... 8 Dispersion Relations ................................................................................................................................. 8 Degenerate Modes ................................................................................................................................. 10 Radiation Properties ............................................................................................................................... 12 Band gaps ................................................................................................................................................ 13 Comparison with Photonic Crystals ........................................................................................................ 16 Plasmonic Crystals with Non-Circular Cross Section Pillars ........................................................................ 17 Applications................................................................................................................................................. 19 Conclusion ................................................................................................................................................... 21 References .................................................................................................................................................. 22
Introduction Plasmonic crystals are periodic two-dimensional surface-relief structures formed on a metal surface, and they support the propagation of surface plasmons. Surface plasmons are quasi two-dimensional electromagnetic waves that propagate along the interface between a metal and a dielectric. The amplitude of the SP decays exponentially with increasing distance into neighbouring media (it is an evanescent wave). The dispersion relation of a plasmonic crystal is similar to photonic crystals such that it also contains band gaps in which no surface plasmons can exist [1]. At the band gap edges, the group velocity is zero, and the surface plasmons form standing waves [2]. The dispersion curves of surface plasmon modes around the lowest band gap are located outside of the light cone, and thus do not couple to radiation [1, 2]. However, higher plasmonic band gaps also exist, and the second surface plasmonic band gap does indeed lie within the light cone and couples to free space radiation [2]. This property makes surface plasmons useful in applications such as light emitting diodes and solar cells. The dispersion relations and radiation properties of one dimensional plasmonic crystals (gratings and corrugations) are relatively simple, and thus this project will focus solely on two-dimensional surface plasmonic crystals. There are several methods to determine the dispersion relations and band structures of a twodimensional plasmonic crystal. The first method was developed by Kretschmann (one of the original scientists who studied surface plasmons in the 60s) and Maradudin, and published nearly 10 years ago [1]. They calculated the dispersion relations of plasmonic crystals composed of two-dimensional hexagonal and square lattices of metallic semielliptical protrusions on a
metal surface, using the βRayleigh-hypothesisβ approximation. However, their method only calculated the dispersion relations outside the light cone, where the radiation properties were left untreated [1]. (They observed the first plasmonic band gap, but not higher ones). This method uses the so called reduced Rayleigh equations for analyzing scatterings. In this case, the approximation was that electromagnetic field within the scattering medium can be ignored, and only the field in the medium of incidence has to be calculated. The original Rayleigh method for scattering yields a pair of coupled integral equations for the scattering amplitudes, whereas this approximation yields only one integral equation [4]. The second method of calculating the dispersion relations of a plasmonic crystal was developed more recently [3]. This method utilizes a semi-analytical rigorous coupled wave analysis (RCWA), also known as Fourier Modal Method The dispersion relations of a two-dimensional hexagonal lattice was determined for both inside and outside the light cone, and the coupling characteristics of the surface plasmons with free space radiations was also studied [2]. The optical (radiation coupling) characteristics and dispersion relations inside the light cone are important for many applications, such as organic LEDs, organic solar cells, and plasmonic band gap lasers [2]. Therefore, this project will focus mainly on the second method (RCWA, Fourier Modal Method) for determining plasmonic crystal dispersion relations.
Theory Rigorous Coupled Wave Analysis (Fourier Modal Method) Let us consider a plasmonic crystal of hexagonal lattice of cylindrical pillars (or holes), shown in Figure 1. The lattice constant is denoted by π¬, the radius of each pillar is π, and the modulation depth is π. The polar and azimuthal angles are denoted by π, and π, respectively. The separation between each lattice point (the more usual βlattice constantβ defined in class) is calculated to be
2π¬ β3
. It is assumed that a linearly polarized monochromatic plane wave is
incident on the surface of the plasmonic crystal.
Figure 1. (a) A positive plasmonic crystal, with an array of pillars protruding from the surface. (b) A negative plasmonic crystal, with an array of air holes.
The dispersion relation of a hexagonal lattice plasmonic crystal with modulation depth that is infinitesimal is shown in Figure 2. In this dispersion relation, an ideal Drude model with plasma frequency ππ is assumed. The reciprocal lattice vector is,
πΎ=
2π π¬ (1)
Figure 2. The dispersion relation of a hexagonal lattice plasmonic crystal of infinitesimal modulation depth, given from the Drude model. The shaded region is where the rigorous coupled wave analysis will be conducted to determine the dispersion relation of finite depth.
In this case where the modulation depth is infinitesimal, there is no band gap. There is only coupling between free space radiation and the surface plasmon. When the modulation depth is finite, then surface plasmons can also couple to other surface plasmons, and band gaps appear [2].
The exact dispersion relation and band structures of the shaded region in Figure 2, with a finite modulation depth, was investigated using rigorous coupled wave analysis (RCWA), also known as the Fourier modal method. The Fourier modal method for two-dimensional plasmonic crystals was originally just an extension of that for one-dimensional gratings. This caused some convergence problems at first [3]. However, a correct formulation for the Fourier modal method for two-dimensional crossed gratings was developed. [3]. To go over the details of this Fourier modal method would be a project in itself, therefore its results will simply be stated. The analysis was conducted using silver as the metal (since in practical photonics applications, silver is often used as the metal layer, due to its low absorption in the visible spectrum) [4]. The dielectric function of silver was retrieved from Johnson and Christy [4]. The lattice constant is π¬ = 600ππ, and the modulation depth is π = 20ππ. The dielectric constant of the ambient is assumed to be the permittivity of free space. A total number of 1681 diffraction orders were used in the calculation, corresponding to 41 diffraction orders in each direction of two reciprocal lattice vectors. This provided enough accuracy, and the calculation time was kept short. When the number of diffraction orders were increased beyond 1681, it was noted that the dispersion relations did not change significantly [2].
Results Dispersion Relations The dispersion relations for the surface plasmon modes were obtained by calculating the absorbance at particular energies as a function of the in-plane wave vector π|| of the incident plane wave. This relationship is plotted in Figure 3 for positive plasmonic crystals of metal pillars for various pillar radii, and in Figure 4 for negative plasmonic crystals of air holes, for various hole radii. The dispersion relation is shown as the curves in the absorbance plots.
Figure 3. The dispersion relations for positive plasmonic crystals, showing absorbance at various energies as a function the in-plane wave vector π|| , for various pillar radii to lattice constant
ratios. Brighter areas indicate higher absorbance (more coupling). Red represents p-polarized incident wave, and blue is s-polarized incident wave.
As shown in Figure 2, in the limit that the modulation depth is infinitesimal, there are four modes in the Ξ-M direction, and three modes in the Ξ-K direction. However, according to the more accurate Fourier modal method calculations, as shown in Figure 3, there are six modes in both directions. The reason being that there was a degeneracy in this region due to infinitesimal modulation depth. In the Ξ-M directions, four modes couple to p-polarized light, and two modes couple to s-polarized light. In the Ξ-K directions, s and p polarized light both couple to three modes. These modes are designated P1 to P7 for p-polarized light coupled modes, and S1 to S5 for s-polarized light coupled modes. As stated earlier, in the infinitesimal modulation depth limit, there are degeneracies, which now can be resolved as the pairs (P2, S1), (P3, S2), (P5, S3), (P6, S4), and (P7, S5). Note that in each pair, one mode is always coupled to the p-polarized light, and the other to s-polarized light. As the pillar radii increase, the shape of the dispersion curves changes slightly, however the number of modes and degeneracy pairs remain the same. The reason that degeneracies occur will be explained in the following section using the wave vector diagrams.
Figure 4. The dispersion relations for the negative plasmonic crystals, for various hole radii to lattice constant ratios. Brightness and colour are defined in the same manner as Figure 3.
Degenerate Modes Figure 5 shows the wave vector diagram in k-space in the first Brillouin zone (the reciprocal lattice of a hexagonal Bravais lattice is another hexagonal lattice that is rotated by 30Β° ). In the first case, the in-plane wave vector π|| is in the Ξ-M direction, where in the latter case it is in the Ξ-K direction.
Figure 5. Wave vector diagrams in k-space, showing the first Brillouin zone, where the in-plane wave vector π|| of the incident radiation is (a) parallel to the Ξ-M direction, and (b) to the Ξ-K direction. Vectors π²π and π²π are the reciprocal lattice vectors. Vectors ππ (π = 1 β¦ 6) are the surface plasmon wave vectors.
The surface plasmon couple to the incident radiation when their wave vectors satisfy the following relationship [2], ππ = π|| + ππ²π + ππ²π (2) Where π and π are integers, and π²π and π²π are the reciprocal lattice vectors of the hexagonal lattice. One may think of this as the incident radiation getting a βboostβ in their wave vector from the two-dimensional gratings, which will allow them to couple to surface plasmon modes. The full set of surface plasmon wave vectors ππ (π = 1 β¦ 6) are noted in green in the diagrams. In particular, for a given incident wave vector π, the surface plasmon wave vectors |ππ | = |ππ |, and |ππ | = |ππ |. Evidently, since the magnitude of their wave vectors are the same, these two
pairs of surface plasmon modes are degenerate when the modulation depth is infinitesimal. However, when the modulation depth is increased to a finite value, the surface plasmon mode of ππ and ππ couple to each other. The wave vector difference between those two modes is, ππ β ππ = βπ²π + ππ²π (3) Which is a linear combination of reciprocal lattice vectors of the hexagonal lattice. The two coupled modes split into two additional modes, an odd mode and an even mode. The odd mode couples to s-polarized radiations (blue curves), and the even mode couples to ppolarized radiations (red curves), and these two modes have different energies [2]. In Figure 5 (b), the incident wave vector is in the Ξ-K direction. Therefore, taking Ξ-K as the axis of symmetry, there are three pairs of degenerate modes in the infinitesimal modulation depth limit. These three pairs are |ππ | = |ππ |, |ππ | = |ππ |, and |ππ | = |ππ |. As the grating depth increases, these pairs of modes will each split off into an s coupled and p coupled modes.
Radiation Properties At the Ξ point (where π = 0 in Figures 3 and 4), some modes are very bright (which translates to strong absorption), and some modes show almost no absorption. This means that some surface plasmon modes can couple to radiation whereas some cannot. As shown in Figures 3, r
for the positive surface plasmon crystals with small pillar radii (Ξ = 0.200), the upper two r
modes are radiative, and the bottom four modes are non-radiative. For larger pillar radii ( Ξ = 0.400), the bottom two modes are radiative, and the upper two are non-radiative. For medium r
r
pillar radii (Ξ = 0.289, or Ξ = 0.305) the middle two modes are radiative, while the rest are
non-radiative. As shown in Figure 4 for negative surface plasmonic crystals, the dependence of the radiation coupling to the size of air holes is exactly opposite to that of positive plasmonic crystals. For smaller hole radii, the bottom two modes are radiative, while for larger hole radii, the top two modes are radiative. Similar investigations into the radiation properties of onedimensional plasmonic crystals have been conducted, most well-known of which are by Barnes, Kitson, and Preist [8, 9].
Band gaps As evident from Figure 3 and 4, part of the dispersion curves for some modes (the non-radiative modes) are missing at the Ξ point. This makes it difficult to calculate the plasmonic band gaps directly. Thus, a method called the Kretschmann geometry was used to determine the band gaps at the Ξ point [2]. Note, the Kretschmann configuration is simply an arrangement of a glass (or another dielectric) prism placed directly against a thin metal layer, on the other side of which is air. Incident radiation goes through the prism, gaining an additional prism momentum which is parallel to the surface (similar to gratings reciprocal lattice vectors, see Equation (2)), strikes the metal surface, and reflects off. This is in contrast to the Otto configuration, in which the prism and the metal are separated by an additional layer of dielectric (usually air). These two configurations are shown in Figure 6.
Figure 6. (a) The Kretschmann configuration. (b) Otto configuration In the particular case studied in this project, the absorbance of internally reflected waves (which are outside the light cone because the modes under study are non-radiative) in terms of energy are calculated, to reveal all the dispersion curves near the Ξ point. Figure 7 (a) shows a schematic diagram of the experimental setup. A silver film of 60 ππ is used as the metallic layer, underneath which is attached a semi-infinite thick glass substrate with index of refraction π = 1.5. A plane wave is incident from the glass side (from underneath). Figure 7 (b) and (c) show the calculated absorbance in terms of energy as a function of pillar radii for the positive plasmonic crystal [2]. Figure 7 (b) shows the results for p-polarized incidence, and (c) for spolarized incidence. Figure 7 (d) and (e) shows the calculated absorbance for a negative plasmonic crystal, and similarly (d) is for p-incidence, and (e) is for s-incidence.
Figure 7. Finding the plasmonic band gaps (a) The Kretschmann configuration for calculating the plasmon modes near the Ξ point. (b) and (c) The calculated absorbance in terms of energy as a function of pillar radii for positive plasmonic crystals, where (b) is for p-incidence, and (c) is for s-incidence. (d) and (e) The absorbance for negative plasmonic crystals, where (d) is for p-incidence and (e) is for s-incidence.
2π
The in-plane wave vector of the incident plane wave was fixed at π|| = ( π¬ , 0), where the dispersion relations of the plasmonic crystal are identical to those with π|| = 0.
π
Evident from the plots in Figure 7, when the ratio Ξ β
0.05, there is only one mode in both the positive and negative plasmonic crystals. As the cylinder radius increases, this mode splits into three modes. Two of the three modes couple to both p and s polarization incidence, whereas one of the three modes only couple to p-polarization incidence. It is also worth to note that for small cylinder radii, no mode couples to s-polarization. The general shape of the energy of the modes as a function of cylinder radii for the negative plasmonic crystal is just the opposite of the positive plasmonic crystal, however the negative crystal modes are less sensitive to cylinder π
radii. The largest band gap for the positive plasmonic crystals is at approximately Ξ β
0.2, with a value of ~0.05 ππ. The largest band gap for the negative plasmonic crystal is at approximately π Ξ
β
0.45, with a value of ~0.03 ππ.
Comparison with Photonic Crystals The band structure of plasmonic crystals are similar to that of two-dimensional photonic crystals or photonic crystal slabs [1, 2]. In fact, the band structures of the two are identical in the limit that the modulation depth becomes infinitesimal. However, in a two-dimensional photonic crystal both TE and TM modes exist, whereas in a plasmonic crystal only TM modes exist [2]. Another major difference is that, at the Ξ point, a plasmonic crystal has three different resonant energies (as shown in Figure 7), whereas in a similar (two-dimensional hexagonal lattice) photonic crystal, there would be four resonant energies (consisting of both TE and TM modes) [2].
Plasmonic Crystals with Non-Circular Cross Section Pillars The dispersion relations and radiation properties of a plasmonic crystal composed of a hexagonal lattice of positive elliptical cylindrical protrusions was investigated. The ellipses have their major axis rotated 15Β° from the x-axis, and their long axis has a length of π =
π¬
, and their
β3
minor axis is π = 0.5π. A single unit cell is shown in Figure 8.
Figure 8. A plasmonic crystal unit cell, with non-circular positive array of pillars. π and π are the major and minor axis, respectively, and π is the azimuthal angle of the plane of incidence measured from the x-axis.
The calculated absorbance in terms of energy for various planes of incidence (by varying incident angle) are plotted in Figure 9. As shown, both the band structure and the radiation properties depend on the azimuthal angle of the plane of incidence. Plots of π = 0Β°, 60Β°, πππ 120Β° correspond to Ξ-M direction, and plots of π = 30Β°, 90Β°, πππ 150Β° correspond
to the Ξ-K direction (see the wave vector diagrams in Figure 5). For plasmonic crystals with perfectly circular pillar arrays, each mode couples exclusively to either p or s polarized incident radiation (when the modulation depth is finite). However for plasmonic crystals with tilted elliptical pillars, some modes can couple to both p and s polarized radiation simultaneously [2].
Figure 9. The dispersion relations for a positive plasmonic crystal with hexagonal lattice of elliptical pillars, with varying azimuthal angle of the plane of incidence.
Applications There are three main important applications of two-dimensional plasmonic crystals. One is the plasmonic band gap laser, in which the surface plasmons are radiated and amplified using a gain medium. The second application is in organic light emitting diode (OLED), surface plasmons at a metallic cathode layer are created by electrically excited excitons, and they radiate out via a surface plasmonic structure that is attached to the metal cathode. The third main application is in organic solar cells, which is almost exactly opposite to that of the OLED. For the plasmonic band gap lasers, the mode density (which is inversely proportional to the group velocity) should be high, in order to lower the lasing threshold [2]. However, at the same time, radiation should be minimized to reduce loss. Therefore, in a plasmonic crystal structure π
similar to the type studied here, the pillar radius should be approximately 0.289 β€ π¬ β€ 0.352, where the upper three modes are degenerate and non-radiative. For the negative plasmonic π
crystals, the hole radius should be approximately π¬ β₯ 0.0.433, where the lower three modes are degenerate and non-radiative. To maximize mode density, the cross section of the pillars and holes should be perfectly circular. For the OLED, surface plasmons should be made to radiate as much as possible. The larger the number of radiative modes, the higher the light extraction (quantum) efficiency is [2]. Based on π
π
this criteria, the optimum radius is π¬ = 0.330 for the positive crystal, and π¬ = 0.352 for the negative crystal. In both cases, there are three radiative modes. An additional application of surface plasmonic crystals is the organic solar cell. The working mechanism of the organic solar cell is almost identically opposite to that of the OLED. An
incident light, after gaining an additional plane parallel wave momentum from a grating, is able to couple to surface plasmon radiative modes. These surface plasmons in term excite excitons, which will increase the light collection efficiency of a solar cell [10]. The same criteria hold in that the surface plasmonic crystal should be made to maximize the number of radiative modes.
Conclusion
A method to investigate and theoretically model the two-dimensional surface plasmonic crystal was developed using the Rigorous Coupled Wave Analysis (Fourier Modal Method). Using this method, the dispersion relations of a two-dimensional hexagonal lattice of protruding cylinders and air holes were studied, as well as the origin and details of their degenerate modes, and their radiation properties. The band gaps near the Ξ point were also determined. They contained several non-radiative modes, therefore a Kretschmann geometry was used to calculate the band gaps. The two-dimensional plasmonic crystal resembles a photonic crystal slab, in the limit that the modulation depth becomes infinitesimal. However, a fundamental difference between the two crystals is that in the former, there are only three modes near the Ξ point, whereas in the latter there exists four modes. The size and shape of the cylindrical pillars and holes seem to also affect the properties of the plasmonic crystal. With a more circular cross section pillars, the mode density increases.
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