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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 4 — Feb. 13, 2012
  • pp: 3663–3674
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Polarizability of nanowires at surfaces: exact solution for general geometry

Jesper Jung and Thomas G. Pedersen  »View Author Affiliations


Optics Express, Vol. 20, Issue 4, pp. 3663-3674 (2012)
http://dx.doi.org/10.1364/OE.20.003663


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Abstract

The polarizability of a nanostructure is an important parameter that determines the optical properties. An exact semi-analytical solution of the electrostatic polarizability of a general geometry consisting of two segments forming a cylinder that can be arbitrarily buried in a substrate is derived using bipolar coordinates, cosine-, and sine-transformations. Based on the presented expressions, we analyze the polarizability of several metal nanowire geometries that are important within plasmonics. Our results provide physical insight into the interplay between the multiple resonances found in the polarizability of metal nanowires at surfaces.

© 2012 OSA

1. Introduction

The electric polarizability of a nanoparticle, i.e. the relative tendency of the electron cloud to be distorted from its normal shape by an external field, is an important concept because it determines the particle’s optical properties. Given the polarizability, it is usually straightforward to determine the light scattering and absorption properties of a nanoparticle [1

1. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

,2

2. L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge2006).

]. By illuminating a metal nanoparticle, collective excitations of the free conduction electrons of the metal can be resonantly excited. Such excitations are known as particle plasmons or localized surface plasmons [3

3. A. V. Zayats and I. I. Smolyaninov, “Near-field photonics: surface plasmon polaritons and localized surface plasmons,” J. Opt. A: Pure Appl. Opt. 5, S16–S50 (2003). [CrossRef]

6

6. S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics 1, 641–648 (2007). [CrossRef]

] and give rise to resonances in the polarizability. Thus, given the polarizability of a metal nanoparticle, its particle plasmons can easily be identified. Particle plasmon resonances are interesting from an application point of view because they allow for strong light scattering and large electromagnetic fields in the vicinity of the particles. These properties suggest a variety of applications, e.g. within surface enhanced raman scattering [7

7. M. Moskovits, “Surface-enhanced spectroscopy,” Rev. Mod. Phys. 57, 783–826 (1985). [CrossRef]

], biomedical detection [8

8. Y. C. Coa, R. Jin, and C. A. Mirkin, “Nanoparticles with Raman spectroscopic fingerprints for DNA and RNA detection,” Science 297, 1536–1540 (2002). [CrossRef]

, 9

9. A. J. Haes and R. P. V. Duyne, “A nanoscale optical biosensor: sensitivity and selectivity of an approach based on the localized surface plasmon resonance spectroscopy of triangular silver nanoparticles,” J. Am. Chem. Soc. 124, 10596–10604 (2002). [CrossRef] [PubMed]

], and plasmon enhanced solar cells [10

10. K. R. Catchpole and A. Polman, “Plasmonic solar cells,” Opt. Express 16, 21793–21800 (2008). [CrossRef] [PubMed]

12

12. V. E. Ferry, J. N. Munday, and H. A. Atwater, “Design considerations for plasmonic photovoltaics,” Adv. Mater. 22, 4794–4808 (2010). [CrossRef] [PubMed]

]. Spherical nanoparticles can be analyzed analytically using the old, but famous Lorenz-Mie scattering theory [13

13. L. Lorenz, “Lysbevægelsen i og udenfor en af plane lysbølger belyst kugle,” K. Dan. Vidensk. Selsk. Skr. 6, 1–62 (1890).

15

15. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

] and, also, the ”two-dimensional” case of a small cylinder has been analyzed a long time ago [16

16. L. Rayleigh, “The dispersal of light by a dielectric cylinder,” Phil. Mag. 36, 365–376 (1918).

, 17

17. J. R. Wait, “Scattering of a plane wave from a circular cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955). [CrossRef]

]. However, for more complicated structures, no analytical solutions can in general be found. In the analysis of such structures, sophisticated numerical schemes such as finite difference time domain [18

18. A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).

], finite element [19

19. J. Jin, The Finite Element Method in Electrodynamics (Wiley, 2002).

], or Green’s function integral equation [20

20. T. Søndergaard, “Modeling of plasmonic nanostructures: Green’s function integral equation methods,” Phys. Status Solidi B 244, 3448–3462 (2007). [CrossRef]

23

23. J. Jung, T. G. Pedersen, T. Søndergaard, K. Pedersen, A. N. Larsen, and B. B. Nielsen, “Electrostatic plasmon resonances of metal nanospheres in layered geometries,” Phys. Rev. B 81, 125413 (2010). [CrossRef]

] approaches are often utilized. However, such approaches are complicated and from a computational point of view often very time-consuming. Thus, analytical modeling, whenever possible, is always favorable.

Fig. 1 (a) cross section of the geometry under consideration. The optical properties are described by the four dielectric constants ε1 to ε4. (b) and (c) two examples of geometries that can be analyzed using the approach.

The paper is organized as follows. In Sec. 2, we present a derivation of the electrostatic polarizability of the geometry sketched in Fig. 1(a). Section 3 presents calculations and an analysis of the polarizability and its resonances for several important geometries. In Sec. 4, we offer our conclusions.

2. Theory

A detailed schematic of the geometry studied is shown in Fig. 2. It consists of four regions 1, 2, 3, and 4. A cylinder of radius r is cut by an infinite interface at an arbitrary position d from its center |d| ≤ r. The segment of the cylinder above (below) the interface is denoted 1 (2) and has optical properties described by ε1 (ε2). The surrounding medium above (below) the interface is denoted 3 (4) and has a dielectric constant ε3 (ε4). To comply with the symmetry of the problem it is an advantage to switch to the bipolar coordinates u and v defined as [34

34. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II (McGraw-Hill Book Company Inc., 1953).

, 35

35. H. E. Lockwood, A Book of Curves (Cambridge University Press, 1963).

]
x=sinhucoshucosvandy=sinvcoshucosv.
(1)
The domains of u and v for the four different regions are shown in Fig. 2. The cut of the interface can be expressed in terms of the angle w, which we, with no loss of generality, choose between 0 < w < π. The surface of the cylinder is defined such that v = w on the part above the interface and v = wπ on the part below the interface. In terms of w, the radius of the cylinder is given as r = 1/sin w and the distance from the center of the cylinder to the interface is given as d = |cot w|. It should be noted that the geometry is defined such that the distance unit is half the intersection of the nanowire by the substrate plane, i.e. half the dashed line in Fig. 2. When investigating a geometry with a specific physical dimension the polarizabilities obtained in the present work should therefore be multiplied by the square of this distance measured in physical units in order to convert the results into standard units.

Fig. 2 Cross section of the geometry analyzed. In bipolar coordinates (u and v), the four different regions in the xy plane are given as ε1 : {−∞ < u < ∞ and w < v < π}, ε2 : {−∞ < u < ∞ and −π < v < wπ}, ε3 : {−∞ < u < ∞ and 0 < v < w}, and ε4 : {−∞ < u < ∞ and wπ < v < 0}.

The total potential ϕi(u,v) is written as a sum of the incident potential ϕ0(u,v) and the scattered potential φi(u,v). We consider the two linear independent polarizations along y and x, vertical and horizontal, respectively. For vertical (horizontal) polarization we look for solutions that are even (odd) functions of x and thereby u. Thus, for vertical polarization we may write the potential using cosine-transformations as
ϕi(u,v)=0ϕi¯(λ,v)cos(λu)dλ,ϕi¯(λ,v)=2π0ϕi(u,v)cos(λu)du,
(3)
and for horizontal polarization we may write using sine-transformations
ϕi(u,v)=0ϕi¯(λ,v)sin(λu)dλ,ϕi¯(λ,v)=2π0ϕi(u,v)sin(λu)du.
(4)
In order to comply with the Laplace equation, the scattered part of the potential in λ,v coordinates may be written as
φi¯(λ,v)=ci(λ)cosh(λv)+si(λ)sinh(λv).
(5)
For vertical and horizontal polarizations the incident potentials are given as
ϕ0(v)(u,v)=sinvcoshucosv{1forv>0ε3ε4forv<0andϕ0(h)(u,v)=sinhucoshucosv,
(6)
respectively. Utilizing the approach presented in Ref. [33

33. J. Jung and T. G. Pedersen, “Exact polarizability and plasmon resonances of partly buried nanowires,” Opt. Express 19, 22775–22785 (2011). [CrossRef] [PubMed]

], it is easy to show that the vertical and horizontal polarizabilities may be found as
α(v)=4π0λs3(λ)dλandα(h)=4π0λc3(λ)dλ.
(7)
The constants s3(λ) and c3(λ) can be derived using the boundary conditions. In order to utilize the boundary conditions for the normal derivative of the potential, we need the following two expressions for the derivative with respect to v of the incident potential. By taking the derivative of Eq. (6) and using Eqs. (3) and (4) it can be shown that
vϕ¯0(v)(λ,v)=2λcosh[λ(π|v|)]sinh[λπ]andvϕ¯0(h)(λ,v)=2λsinh[λ(π|v|)]sinh(λπ)sgn(v).
(8)
With the notation Cpq ≡ cosh[λ(qw)] and Spq ≡ sinh[λ (qw)] the continuity of the potential for both polarizations yields
c1(λ)C01s1(λ)S01=c3(λ)C01s3(λ)S01,c2(λ)C11s2(λ)S11=c4(λ)C11s4(λ)S11,c1(λ)C10+s1(λ)S10=c2(λ)C10s2(λ)S10,c3(λ)=c4(λ).
(9)
For vertical polarization the boundary conditions for the normal derivative of the potential yields
ε1[c1(λ)S01+s1(λ)C01+2C11S10]=ε3[c3(λ)S01+s3(λ)C01+2C11S10],ε2[c2(λ)S11+s2(λ)C11+ε3ε42C01S10]=ε4[c4(λ)S11+s4(λ)C11+ε3ε42C01S10],ε1[c1(λ)S10+s1(λ)C10+2S10]=ε2[c2(λ)S10+s2(λ)C10+ε3ε42S10],ε3s3(λ)=ε4s4(λ),
(10)
and for horizontal polarization they read
ε1[c1(λ)S01+s1(λ)C01+2S11S10]=ε3[c3(λ)S01+s3(λ)C01+2S11S10],ε2[c2(λ)S11+s2(λ)C11+2S01S10]=ε4[c4(λ)S11+s4(λ)C11+2S01S10],ε1[c1(λ)S10+s1(λ)C10]=ε2[c2(λ)S10+s2(λ)C10],ε3s3(λ)=ε4s4(λ).
(11)
By solving the equation system formed by the 8 equations of Eqs. (9) and (10) [Eqs. (9) and (11)], s3(λ) [c3(λ)] can be found and the polarizability can be calculated using Eq. (7). Generally both the solution of s3(λ) and c3(λ) are fractions on the form 2N/(DS10), where D and N are given as
D=p,qDpqCpqandN=p,qNpq{CpqC10}
(12)
with
D20=(ε1+ε2)(ε1+ε3)(ε2+ε4)(ε3+ε4),D24=(ε1ε2)(ε1ε3)(ε2ε4)(ε3ε4),D22=2(ε12ε2ε3)(ε2ε3ε42)2ε1(ε2ε3)2ε4,D02=2(ε12+ε2ε3)(ε2ε3+ε42)+2ε1(ε2+ε3)2ε4,D00=2(ε12ε2ε3)(ε2ε3ε42)2ε1(ε2+ε3)2ε48ε1ε2ε3ε4.
(13)
For s3(λ), i.e. vertical polarization, N is computed from
N14=(ε1ε2)(ε1ε3)(ε2ε4)ε3,N34=(ε1ε2)(ε1ε3)(ε2ε4)ε4,N32=(ε1+ε2)(ε1ε3)(ε2+ε4)ε4,N12=(ε1+ε2)(ε1+ε3)(ε2ε4)ε3,N12=2(ε12ε42+ε22ε32)+(ε3+ε4)[ε12(ε4ε2)+ε22(ε3ε1)+(ε1+ε2)ε3ε4]ε1ε2(ε32+6ε3ε4+ε42).
(14)
There exists a simple symmetry between s3(λ) and c3(λ) and thereby also α(v) and α(h): By substituting εi with 1/εi for all i s3(λ) transforms intoc3(λ) and vice versa. In the following, we will therefore mainly focus on vertical polarization, because the corresponding results for horizontal polarization can be obtained by performing simple substitutions.

The general expression for s3(λ) simplifies substantially for simpler geometries. For a cut cylinder in a homogenous surrounding (ε1ε and ε2 = ε3 = ε4εh) s3(λ), for example, reduces to
s3(λ)=N12(C12C10)+N32(C32C10)D00+D20C20+D02C022S10,
(15)
where D00 = −4εεh, D20 = (ε + εh)2, D02 = −(εεh)2, N12 = (ε − 3εh)(εεh)/2, and N32 = (ε + εh)(εεh)/2.

For w → 0 the geometry describes a full cylinder lying on a substrate. In this case, the dominant contributions to the integrals of Eq. (7) are from λ ≫ 1 and we are therefore allowed to approximate as C11S11 ≈ exp[λ (πw)]/2 and C10S10 ≈ exp(λπ)/2. Using this and taking ε2 = ε4 we find
s3(λ)2(ε1ε3)ε4eλ(2π+w)[e2λπ(ε1+ε4)2(e2λw1)(ε1ε3)](ε1+ε4)[ε3(ε1+ε4)cosh(λw)+(ε32+ε1ε4)sinh(λw)],
(16)
which for λ ≫ 1 can be approximated as
s3(λ)2(ε1ε3)ε4eλwε3(ε1+ε4)cosh(λw)+(ε32+ε1ε4)sinh(λw).
(17)
By performing the integral of Eq. (7) using the above expression for s3(λ) we obtain the simple result for the vertical polarizability
α(v)=4πε4w2(ε4ε3)Li2[(ε3ε1)(ε3ε4)(ε3+ε1)(ε3+ε4)],
(18)
where Lis(z) is the polylogarithm (or Jonquire’s function [36

36. A. Jonquiere, “Note sur la serie n=1xnns,” Bulletin de la Socit Mathmatique de France 17, 142–152 (1889).

]). A resonance at ε1 = −ε3 is revealed. Note that, as expected, the polarizability is proportional to the radius squared r2 = 1/sin2 w ≈ 1/w2 for w small. By performing similar approximations for the horizontally polarized case c3(λ) can be found as
c3(λ)2(ε1ε3)ε3exp(λw)(ε1+ε4)ε3cosh(λw)+(ε32+ε1ε4)sinh(λw),
(19)
which integrates to
α(h)=4πε3w2(ε4ε3)Li2[(ε3ε1)(ε3ε4)(ε3+ε1)(ε3+ε4)].
(20)
Again a resonance at ε1 = −ε3 is revealed. It should be noted that because of the approximations utilized C11S11 ≈ exp[λ (πw)]/2 and C10S10 ≈ exp(λπ)/2 the substitutional symmetry between s3(λ) and c3(λ) has disappeared. Eq. (19) cannot be obtained from Eq. (17) by utilizing the simple symmetry stated above. Our results show that, for a cylinder lying on a surface, it is the dielectric constant of the medium above the surface that dictates the resonances of the polarizability. In fact, the resonance condition ε1 = −ε3 is identical to that of a cylinder in a homogenous ε3 surrounding. However, the scaling of the polarizability for the two polarizations are different. For vertical resonances it scales with ε4 whereas for horizontal resonances it scales with ε3.

By Taylor expanding the general expression for s3(λ) [Eqs. (12), (14), and (15)] for λ small we find
s3(λ)2λ[1π+ε31+ε41(ε11+ε41)(πw)+(ε21+ε31)w].
(21)
From this expansion it can be seen that s3(λ), and therefore also α(v), has a resonance at the condition
(ε11+ε41)(πw)+(ε21+ε31)w=0.
(22)
Note the dependence on w of the resonance condition. For horizontal polarization the resonance condition can be obtained by utilizing the simple symmetry εi → 1/εi. This yields
(ε1+ε4)(πw)+(ε2+ε3)w=0.
(23)

In the general case, for an arbitrary w, the integrals of Eq. (7) cannot be performed analytically. It is only possible in a few special cases. For a half-buried cylinder w = π/2 the results are presented in Ref. [33

33. J. Jung and T. G. Pedersen, “Exact polarizability and plasmon resonances of partly buried nanowires,” Opt. Express 19, 22775–22785 (2011). [CrossRef] [PubMed]

], but e.g. also for w = 2π/3, corresponding to a 3/4 buried cylinder, an analytical result can be obtained. We will not present these rather comprehensive calculations here, but instead note that when the analytical expressions of s3(λ) and c3(λ) are given, the integrals of Eq. (7) are straightforward to evaluate using numerical integration. There are no problems with convergence or singularities. In fact, if the integrals over λ are taken from 0 to 10 a fully converged result is obtained, except if w is very small.

3. Results

First, we present calculations of the polarizability of three different configurations using the approach outlined in Sec. 2. We consider (a) a cut cylinder in a homogenous surrounding (ε1 = ε and ε2 = ε3 = ε4 = εh = 1), (b) a cut cylinder on a quartz surface (ε1 = ε, ε2 = ε4 = εh = 2.25, and ε3 = 1), and (c) a full cylinder partly buried in a quartz substrate (ε1 = ε2 = ε, ε4 = 2.25, and ε3 = 1). In order to identify the resonances, we consider metal-like cylinders with complex dielectric constants that have negative real parts and a small imaginary part. The small imaginary part has been added to prevent singularities at the resonances of the polarizability. Thus, the dielectric constant of the cylinder is in the following calculations given by ε = εr + 0.01i, where εr is negative. The result for configuration (a) is presented in Fig. 3. The figures to the left display the real (top) and imaginary (bottom) part of the vertical polarizability. The figures to the right display the same, but for the horizontal polarizability. Three different w’s are considered: w = π/3 corresponding to a 3/4 cylinder, w = π/2 a half cylinder, and w = 2π/3 a 1/4 cylinder. For configuration (a) Eqs. (22) and (23) give the following resonance conditions (for vertical and horizontal polarization, respectively)
ε=εhπwπ+wandε=εhπ+wπw.
(24)
For vertical polarization the three different w’s considered give resonances at εr = −1/2, −1/3, and −1/5, and for horizontal polarization we find resonances at εr = −2, −3, and −5. All these resonances are clearly seen both in the real and the imaginary part of the polarizability (Fig. 3). For w small a resonance at εr = −εh = −1 is also expected [c.f. Eqs. (18) and (20)]. For w = π/3 this resonance is visible in the polarizability of both polarizations (Fig. 3). From Fig. 3 it can be seen that the resonance in the vertical polarizability moves towards larger εr for an increasing angle w, whereas the resonance in the horizontal polarizability moves towards smaller εr. For ordinary plasmonic metals like silver and gold this corresponds to a blue-shift of the resonance in the vertical polarizability and a red-shift in the horizontal. From the presented results it is clear that it is the geometry of the nanowire that dictates the location of the resonance. In fact it is the ratio between the size of the nanowire along the induced dipole moment to the size of the nanowire perpendicular to the induced dipole moment that is important. Thus, for the vertical polarizability it is the height to the width ratio, which is slowly decreasing for an increasing w that accounts for the small blue-shift of the resonance. For the horizontal polarizability it is the width to the height ratio, which is strongly increasing with larger w that accounts for the large red-shift of the resonance.

Fig. 3 Polarizability as a function of εr for three differently cut cylinders.

For a cut cylinder on a surface, configuration (b), the calculated polarizability is depicted in Fig. 4. For this configuration the resonance conditions from Eqs. (22) and (23) give
ε=εhε3πwε3π+εhwandε=εhπ+ε3wπw.
(25)
For the three w’s and vertical polarization this yields εr ≈ −0.86,−0.53, and εr = −0.3. For horizontal polarization we expect resonances at εr = −3.875, −5.5, and −8.75. These resonances are all clearly seen in the calculated polarizability (Fig. 4). For configuration (b) the resonance at ε = −ε3 = −1, which is expected for w small [c.f. Eqs. (18) and (20)], is largely visible in both the vertical and horizontal polarizability for w = π/3. Again it can be seen how the resonance in the vertical (horizontal) polarizability blue- (red)-shifts when the angle w increases. As described above, this is due to the changing geometry of the nanowire when w increases. By comparing Fig. 4 to Fig. 3 is can be seen that all the resonances (except for the one fixed at εr = −1 in the w = π/3 configuration) are red-shifted. This is because the effective surrounding index that the nanowire feels in the geometry of Fig. 4 is larger than in the geometry of Fig. 3. However, it should be noted that the relative shift of the resonances is very similar in the two configurations.

Fig. 4 Polarizability as a function of εr for three differently cut cylinders on a quartz surface.

The polarizability of configuration (c), a partly buried full cylinder, is presented in Fig. 5. In this case, Eqs. (22) and (23) yield
ε=πε3ε4ε3(πw)+ε4wandε=ε4(πw)+ε3wπ.
(26)
Thus, for the three w’s considered, we expect resonances at εr ≈ −1.59, −1.38, and −1.23 and εr ≈ −1.83, −1.63, and −1.42 for vertical and horizontal polarization, respectively. All these resonances are clearly seen in the polarizability (Fig. 5). However, the figure reveals that three different resonances come into play: The one determined by Eq. (26), and two others fixed at εr = −ε3 = −1 and εr = −εh = −2.25. The latter two are identical to the resonances of a full cylinder in a homogenous surrounding described by dielectric constants εh = 1 and εh = 2.25, respectively. It is clear from the results that the εr = −1 resonance is strongest when w is small, i.e when a large part of the cylinder is in the medium 3, whereas the εr = −2.25 resonance is strongest for w large, i.e. when the cylinder is largely buried in the substrate. These observations agree nicely with the theory for w very small, which predicts a resonance at ε1 = −ε3 = −1, see Eqs. (18) and (20). From the results, in particular the imaginary part of the polarizability, it can also be seen how the effective index of the surrounding medium controls the cutoff between absorption in the blue and the red part of the dielectric spectrum. The full range of absorption is restricted to the dielectric range εr ∈ [−2.25,−1] with a gap separating the blue and the red parts. The cutoff controlling the position of this gap clearly reflects the degree of burial of the nanowire. Thus, when most of the nanowire is in air, significant absorption in the blue part of the spectrum is observed. However, as the nanowire moves into the substrate, the large absorption shifts towards the red part of the spectrum. The main results of the three configurations are summarized in Table 1.

Fig. 5 Polarizability as a function of εr for three cylinders differently buried in a quartz surface.

Table 1. Summary of the resonance conditions for the nanowire dielectric constant of the three configurations (a), (b), and (c) investigated.

table-icon
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Lastly, we have calculated the polarizability in the case of w → 0 taking ε2 = ε4 = εh, i.e the limit where the geometry resembles a full cylinder lying on a surface. This allows us to use the approximate analytical expressions of Eqs. (18) and (20) to calculate the polarizability. First, we consider a cylinder with a metal-like dielectric constant ε = εr + 0.01i, where εr is negative. For ε3 = 1 and three different substrates εh = 2.25 (quartz), 5 (silicon nitride), and 11.9 (silicon), the imaginary part of the horizontal polarizability is displayed in Fig. 6 (a). Because the polarizabilities for the two polarizations in this configuration, except for a scaling, are identical, we only present results for the horizontal polarizability. The results show that the imaginary part of the polarizability rises at the resonance εr = −1 and vanishes again for εr < −εh. This is clearly seen from the zoom [inset of Fig. 6 (a)]. Thus, by choosing a substrate with a large dielectric constant, absorption of light in the nanowire is sustained over a relatively broad range of wavelengths. This is more clearly seen in Fig. 6 (b), where we have calculated the imaginary part of the horizontal polarizability versus the photon energy of a silver nanowire on different substrates using a dielectric constant for silver taken from the experiments of Johnson and Christy [37

37. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]

]. Note how the low-frequency tail of the resonance broadens as the dielectric constant of the substrate increases. On the high-frequency side of the resonance the effect of interband absorption in the silver is seen. Lastly it should be noted, that nanocylinders on surfaces have been analyzed before using numerical analysis [38

38. J. R. Arias-Gonzalez and M. Nieto-Vesperinas, “Resonant near-field eigenmodes of nanocylinders on flat surfaces under both homogenous and inhomogenous lightwave excitation,” J. Opt. Soc. Am. A 18, 657–665 (2001). [CrossRef]

] and more recently transformation optics [39

39. A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Plasmonic hybridization between nanowires and a metallic surface: a transformation optics approach,” ACS Nano 53293–3308 (2011). [CrossRef] [PubMed]

]. Compared to the full approaches of Refs. [38

38. J. R. Arias-Gonzalez and M. Nieto-Vesperinas, “Resonant near-field eigenmodes of nanocylinders on flat surfaces under both homogenous and inhomogenous lightwave excitation,” J. Opt. Soc. Am. A 18, 657–665 (2001). [CrossRef]

, 39

39. A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Plasmonic hybridization between nanowires and a metallic surface: a transformation optics approach,” ACS Nano 53293–3308 (2011). [CrossRef] [PubMed]

] the approximate polarizabilities of Eqs. (18) and (20) are easy to evaluate in that they only involve the calculation of a single polylogarithm.

Fig. 6 Imaginary part of the horizontal polarizability for a full cylinder lying on three different surfaces. (a) versus εr and (b) versus the photon energy. In (a), we use ε = εr + 0.01i and in (b) we use a dielectric constant for silver taken from the experiments of Ref. [37].

4. Conclusion

The electrostatic polarizability of a general geometry consisting of two nanowire segments forming a cylinder that can be arbitrarily buried in a substrate is derived in a semi-analytical approach using bipolar coordinates, cosine-, and sine-transformations. From the derived expressions we have calculated the polarizability of four important metal nanowire geometries: (1) a cut cylinder in a homogenous surrounding, (2) a cut cylinder on a surface, (3) a full cylinder partly buried in a substrate, and (4) a cylinder lying on a surface. Our results give physical insight into the interplay between the multiple resonances of the polarizability of metal nanowire geometries at surfaces, and provide an exact, fast, and easy scheme for optimizing metal nanowire structures for various applications within plasmonics.

Acknowledgments

The authors gratefully acknowledge support from the project “Localized-surface plasmons and silicon thin-film solar cells - PLATOS” financed by the Villum foundation.

References and links

1.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

2.

L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge2006).

3.

A. V. Zayats and I. I. Smolyaninov, “Near-field photonics: surface plasmon polaritons and localized surface plasmons,” J. Opt. A: Pure Appl. Opt. 5, S16–S50 (2003). [CrossRef]

4.

S. A. Maier and H. A. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005). [CrossRef]

5.

W. A. Murray and W. L. Barnes, “Plasmonic materials,” Adv. Mater. 19, 3771–3782 (2007). [CrossRef]

6.

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics 1, 641–648 (2007). [CrossRef]

7.

M. Moskovits, “Surface-enhanced spectroscopy,” Rev. Mod. Phys. 57, 783–826 (1985). [CrossRef]

8.

Y. C. Coa, R. Jin, and C. A. Mirkin, “Nanoparticles with Raman spectroscopic fingerprints for DNA and RNA detection,” Science 297, 1536–1540 (2002). [CrossRef]

9.

A. J. Haes and R. P. V. Duyne, “A nanoscale optical biosensor: sensitivity and selectivity of an approach based on the localized surface plasmon resonance spectroscopy of triangular silver nanoparticles,” J. Am. Chem. Soc. 124, 10596–10604 (2002). [CrossRef] [PubMed]

10.

K. R. Catchpole and A. Polman, “Plasmonic solar cells,” Opt. Express 16, 21793–21800 (2008). [CrossRef] [PubMed]

11.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010). [CrossRef] [PubMed]

12.

V. E. Ferry, J. N. Munday, and H. A. Atwater, “Design considerations for plasmonic photovoltaics,” Adv. Mater. 22, 4794–4808 (2010). [CrossRef] [PubMed]

13.

L. Lorenz, “Lysbevægelsen i og udenfor en af plane lysbølger belyst kugle,” K. Dan. Vidensk. Selsk. Skr. 6, 1–62 (1890).

14.

G. Mie, “Beitrage zur optik truber medien speziell kolloidaler metallosungen,” Ann. Physik. 330, 337–445 (1908). [CrossRef]

15.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

16.

L. Rayleigh, “The dispersal of light by a dielectric cylinder,” Phil. Mag. 36, 365–376 (1918).

17.

J. R. Wait, “Scattering of a plane wave from a circular cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955). [CrossRef]

18.

A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).

19.

J. Jin, The Finite Element Method in Electrodynamics (Wiley, 2002).

20.

T. Søndergaard, “Modeling of plasmonic nanostructures: Green’s function integral equation methods,” Phys. Status Solidi B 244, 3448–3462 (2007). [CrossRef]

21.

O. J. F. Martin, C. Girard, and A. Dereux, “Generalized field propagator for electromagnetic scattering and light confinement,” Phys. Rev. Lett. 74, 526–529 (1995). [CrossRef] [PubMed]

22.

F. J. Garcia de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogenous dielectrics,” Phys. Rev. B 65, 115418 (2002). [CrossRef]

23.

J. Jung, T. G. Pedersen, T. Søndergaard, K. Pedersen, A. N. Larsen, and B. B. Nielsen, “Electrostatic plasmon resonances of metal nanospheres in layered geometries,” Phys. Rev. B 81, 125413 (2010). [CrossRef]

24.

F. Hallermann, C. Rockstuhl, S. Fahr, G. Seifert, S. Wackerow, H. Graener, G. V. Plessen, and F. Lederer, “On the use of localized plasmon polaritons in solar cells,” Phys. Status Solidi A 205, 2844–2861 (2008). [CrossRef]

25.

J. W. Yoon, W. J. Park, K. J. Lee, S. H. Song, and R. Magnusson, “Surface-plasmon mediated total absorption of light into silicon,” Opt. Express 19, 20673–20680 (2011). [CrossRef] [PubMed]

26.

P. C. Waterman, “Surface fields and the T matrix,” J. Opt. Soc. Am. A 16, 2968–2977 (1999). [CrossRef]

27.

A. V. Radchik, A. V. Paley, G. B. Smith, and A. V. Vagov, “Polarization and resonant absorption in intersecting cylinders and spheres,” J. Appl. Phys. 76, 4827–4835 (1994). [CrossRef]

28.

A. Salandrino, A. Alu, and N. Engheta, “Parallel, series, and intermediate interconnects of optical nanocircuit elements. 1. Analytical solution,” J. Opt. Soc. Am. B 24, 3007–3013 (2007). [CrossRef]

29.

M Pitkonen, “A closed-form solution for the polarizability of a dielectric double half-cylinder,” J. Electromagn. Waves Appl. 24, 1267–1277 (2010). [CrossRef]

30.

H. Kettunen, H. Wallen, and A. Sihvola, “Polarizability of a dielectic hemisphere,” J. Appl. Phys. 102, 044105 (2007). [CrossRef]

31.

Y. Luo, J. B. Pendry, and A. Aubry, “Surface plasmons and singularities,” Nano Lett. 10, 4186–4191 (2010). [CrossRef]

32.

Y. Luo, A. Aubry, and J. B. Pendry, “Electromagnetic contribution to surface-enhanced Raman scattering from rough metal surfaces: a transformation optics approach,” Phys. Rev. B 83, 155422 (2011). [CrossRef]

33.

J. Jung and T. G. Pedersen, “Exact polarizability and plasmon resonances of partly buried nanowires,” Opt. Express 19, 22775–22785 (2011). [CrossRef] [PubMed]

34.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II (McGraw-Hill Book Company Inc., 1953).

35.

H. E. Lockwood, A Book of Curves (Cambridge University Press, 1963).

36.

A. Jonquiere, “Note sur la serie n=1xnns,” Bulletin de la Socit Mathmatique de France 17, 142–152 (1889).

37.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]

38.

J. R. Arias-Gonzalez and M. Nieto-Vesperinas, “Resonant near-field eigenmodes of nanocylinders on flat surfaces under both homogenous and inhomogenous lightwave excitation,” J. Opt. Soc. Am. A 18, 657–665 (2001). [CrossRef]

39.

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Plasmonic hybridization between nanowires and a metallic surface: a transformation optics approach,” ACS Nano 53293–3308 (2011). [CrossRef] [PubMed]

OCIS Codes
(000.3860) General : Mathematical methods in physics
(230.5750) Optical devices : Resonators
(240.6680) Optics at surfaces : Surface plasmons

ToC Category:
Optics at Surfaces

History
Original Manuscript: October 28, 2011
Revised Manuscript: December 12, 2011
Manuscript Accepted: January 27, 2012
Published: January 31, 2012

Citation
Jesper Jung and Thomas G. Pedersen, "Polarizability of nanowires at surfaces: exact solution for general geometry," Opt. Express 20, 3663-3674 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-3663


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References

  1. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).
  2. L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge2006).
  3. A. V. Zayats and I. I. Smolyaninov, “Near-field photonics: surface plasmon polaritons and localized surface plasmons,” J. Opt. A: Pure Appl. Opt.5, S16–S50 (2003). [CrossRef]
  4. S. A. Maier and H. A. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys.98, 011101 (2005). [CrossRef]
  5. W. A. Murray and W. L. Barnes, “Plasmonic materials,” Adv. Mater.19, 3771–3782 (2007). [CrossRef]
  6. S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics1, 641–648 (2007). [CrossRef]
  7. M. Moskovits, “Surface-enhanced spectroscopy,” Rev. Mod. Phys.57, 783–826 (1985). [CrossRef]
  8. Y. C. Coa, R. Jin, and C. A. Mirkin, “Nanoparticles with Raman spectroscopic fingerprints for DNA and RNA detection,” Science297, 1536–1540 (2002). [CrossRef]
  9. A. J. Haes and R. P. V. Duyne, “A nanoscale optical biosensor: sensitivity and selectivity of an approach based on the localized surface plasmon resonance spectroscopy of triangular silver nanoparticles,” J. Am. Chem. Soc.124, 10596–10604 (2002). [CrossRef] [PubMed]
  10. K. R. Catchpole and A. Polman, “Plasmonic solar cells,” Opt. Express16, 21793–21800 (2008). [CrossRef] [PubMed]
  11. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater.9, 205–213 (2010). [CrossRef] [PubMed]
  12. V. E. Ferry, J. N. Munday, and H. A. Atwater, “Design considerations for plasmonic photovoltaics,” Adv. Mater.22, 4794–4808 (2010). [CrossRef] [PubMed]
  13. L. Lorenz, “Lysbevægelsen i og udenfor en af plane lysbølger belyst kugle,” K. Dan. Vidensk. Selsk. Skr.6, 1–62 (1890).
  14. G. Mie, “Beitrage zur optik truber medien speziell kolloidaler metallosungen,” Ann. Physik.330, 337–445 (1908). [CrossRef]
  15. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  16. L. Rayleigh, “The dispersal of light by a dielectric cylinder,” Phil. Mag.36, 365–376 (1918).
  17. J. R. Wait, “Scattering of a plane wave from a circular cylinder at oblique incidence,” Can. J. Phys.33, 189–195 (1955). [CrossRef]
  18. A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).
  19. J. Jin, The Finite Element Method in Electrodynamics (Wiley, 2002).
  20. T. Søndergaard, “Modeling of plasmonic nanostructures: Green’s function integral equation methods,” Phys. Status Solidi B244, 3448–3462 (2007). [CrossRef]
  21. O. J. F. Martin, C. Girard, and A. Dereux, “Generalized field propagator for electromagnetic scattering and light confinement,” Phys. Rev. Lett.74, 526–529 (1995). [CrossRef] [PubMed]
  22. F. J. Garcia de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogenous dielectrics,” Phys. Rev. B65, 115418 (2002). [CrossRef]
  23. J. Jung, T. G. Pedersen, T. Søndergaard, K. Pedersen, A. N. Larsen, and B. B. Nielsen, “Electrostatic plasmon resonances of metal nanospheres in layered geometries,” Phys. Rev. B81, 125413 (2010). [CrossRef]
  24. F. Hallermann, C. Rockstuhl, S. Fahr, G. Seifert, S. Wackerow, H. Graener, G. V. Plessen, and F. Lederer, “On the use of localized plasmon polaritons in solar cells,” Phys. Status Solidi A205, 2844–2861 (2008). [CrossRef]
  25. J. W. Yoon, W. J. Park, K. J. Lee, S. H. Song, and R. Magnusson, “Surface-plasmon mediated total absorption of light into silicon,” Opt. Express19, 20673–20680 (2011). [CrossRef] [PubMed]
  26. P. C. Waterman, “Surface fields and the T matrix,” J. Opt. Soc. Am. A16, 2968–2977 (1999). [CrossRef]
  27. A. V. Radchik, A. V. Paley, G. B. Smith, and A. V. Vagov, “Polarization and resonant absorption in intersecting cylinders and spheres,” J. Appl. Phys.76, 4827–4835 (1994). [CrossRef]
  28. A. Salandrino, A. Alu, and N. Engheta, “Parallel, series, and intermediate interconnects of optical nanocircuit elements. 1. Analytical solution,” J. Opt. Soc. Am. B24, 3007–3013 (2007). [CrossRef]
  29. M Pitkonen, “A closed-form solution for the polarizability of a dielectric double half-cylinder,” J. Electromagn. Waves Appl.24, 1267–1277 (2010). [CrossRef]
  30. H. Kettunen, H. Wallen, and A. Sihvola, “Polarizability of a dielectic hemisphere,” J. Appl. Phys.102, 044105 (2007). [CrossRef]
  31. Y. Luo, J. B. Pendry, and A. Aubry, “Surface plasmons and singularities,” Nano Lett.10, 4186–4191 (2010). [CrossRef]
  32. Y. Luo, A. Aubry, and J. B. Pendry, “Electromagnetic contribution to surface-enhanced Raman scattering from rough metal surfaces: a transformation optics approach,” Phys. Rev. B83, 155422 (2011). [CrossRef]
  33. J. Jung and T. G. Pedersen, “Exact polarizability and plasmon resonances of partly buried nanowires,” Opt. Express19, 22775–22785 (2011). [CrossRef] [PubMed]
  34. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II (McGraw-Hill Book Company Inc., 1953).
  35. H. E. Lockwood, A Book of Curves (Cambridge University Press, 1963).
  36. A. Jonquiere, “Note sur la serie ∑n=1∞xnns,” Bulletin de la Socit Mathmatique de France17, 142–152 (1889).
  37. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972). [CrossRef]
  38. J. R. Arias-Gonzalez and M. Nieto-Vesperinas, “Resonant near-field eigenmodes of nanocylinders on flat surfaces under both homogenous and inhomogenous lightwave excitation,” J. Opt. Soc. Am. A18, 657–665 (2001). [CrossRef]
  39. A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Plasmonic hybridization between nanowires and a metallic surface: a transformation optics approach,” ACS Nano53293–3308 (2011). [CrossRef] [PubMed]

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