Complex band structure and plasmon lattice Green s function of a periodic metal-nanoparticle chain

Transcription

1 Coplex band structure and plason lattice Green s function of a periodic etal-nanoparticle chain Kin Hung Fung, Ross Chin Hang Tang, and C. T. Chan Departent of Physics, The Hong Kong University of Science and Technology, Hong Kong, China When the surface plason resonance in a etal-nanoparticle chain is excited at one point, the response signal will generally decay down the chain due to absorption and radiation losses. The decay length is a key paraeter in such plasonic systes. By studying the plason lattice Green s function, we found that the decay length is generally governed by two exponential decay constants with phase factors corresponding to guided Bloch odes and one power-law decay with a phase factor corresponding to that of free space photons. The results show a high level of siilarity between the absorptive and radiative decay channels. By analyzing the poles (and the corresponding residues) of the Green s function in a transfored coplex reciprocal space, the doinant decay channel of the real-space Green s function is understood. PACS nuber(s): Bf, 73..Lp, 73..Mf, g I. INTRODUCTION Band theoretic techniques are widely applied in describing the eigenodes of periodic systes. With the help of the Bloch s theore, a band structure for propagating odes in a lossless periodic syste is usually displayed as a plot of the eigenode as a function

2 of real frequency (ω ) versus real Bloch wave vector (k ) in the reduced-zone schee. Such a real-k band-structure description has been generalized for including evanescent waves within bandgap by allowing the Bloch wave vectors to be coplex quantities. 3 4 This generalized coplex- k description is soeties called the coplex-bandstructure description. There are several advantages of considering the coplex band structure. For exaple, the agnitude of the iaginary part of the k -vector gives the strength and the physical nature of the bandgaps. In addition, it can describe the decaying waves in lossy systes. It is generally believed that a non-zero iaginary part of k iplies an exponential decay (or growth) of the wave with a decay constant given by = /I(. Although the actual decay profile ay depend on the incident field of a particular proble, such kind of exponential decay characteristics are indeed ebedded in the Green s function. 5 6 For exaple, the frequency-doain Green s function for a hoogeneous one-diensional (D) wave equation takes the for, 5 Re( ) I( ) G( ω, x)~ e i k x e k x, where ω is the excitation frequency, k is the corresponding coplex wave nuber, and x is the spatial coordinate. Since the Green s function contains all the essential inforation on wave propagation, we can have a useful interpretation of the coplex wave properties by studying the decay profile of the Green s function and its relation to the coplex band structure. Although there were soe studies in the literature that is concerned with the Green s function for inhoogeneous lossy edia, only a few of the focus on surface plasons, 7 8 which have been shown to exhibit any interesting properties in recent years. 9 In

3 plasonic systes, which are intrinsically lossy due to absorption and radiation, the priary concern is the decay length of the plasonic odes that can be excited in a certain point. In this paper, we study the Green s function for the plasonic waves that propagate along a linear array of etal nanoparticles (MNPs) We will first discuss the nuerical results on the coplex band structure (Sec. II) and the lattice Green s function (Sec. III) for such MNP array. Then, we will propose an ansatz which gives a closed-for expression for the lattice Green s function and we will show that the ansatz is indeed the correct solution by considering a generating function for the lattice Green s function (Sec. IV). II. COMPLEX BAND STRUCTURE We consider a linear array of identical MNPs (with lattice constant a and particle radius r ). For a r 3 and frequencies close to the Fröhlich frequency, 6 the electroagnetic odes of the MNPs can be accurately described by the coupled-dipole equations, 7 ext p = α E + W( R R n ) p n, () n where p is the electric dipole oent of the th particle, α is the single-particle dynaic polarizability, and E ext is the external electric field acting on the th particle. The dynaic propagator, W (r), for a dipole in a hoogeneous host ediu can be written as 7 W (r) = kh [ A( khr) δ uv + B( khr) ru rv / r ], where k h = ω / ch, c h is the speed of uv light in the host ediu, (x) 3 3 A ( x + ix x ) e ix 3 =, B (x) = ( x 3ix + 3x ) e ix, and 3

4 u, v =,, 3 are coponent indices in Cartesian coordinates. Equation () can be written as M unv pnv = n, v E ext u, () where p u and ext E u are the u th coponents of the p and ext E, respectively. For a linear array of MNPs, Eq. () can be further split into decoupled equations in the for, n M p = E n n ext, for either the transverse odes (with p perpendicular to the chain axis) or longitudinal odes ( p parallel to the chain axis). Since the transverse odes have richer properties in its band structure (such as negative slope 3 and strong coupling to free photon 7 ), we will focus on the transverse ode. ext The band structure can be calculated by setting E = and finding pairs of ω and k so that Eq. () has non-trivial Bloch solutions of the for ika p = p e. More explicitly, we have to find the solutions of the equation, 4 a 3 / α ( ω) κ( ω, =, (3) κ = k a Σ + ikhaσ Σ3 where ( ω, i ) h for the transverse ode, Σ n ( ( kh k = Li a n e ) i( kh + k ) a + Li ( e ), and Li n (x) is the polylogarith function of order n. For lossless systes, n ω and k are real nubers. Since the syste we consider here has both radiation loss and absorption loss, real values of ω and k cannot satisfy Eq. (3). There are several ways to define the band structure of lossy systes but here we only focus on the coplex band structure calculated by finding the coplex k that corresponds to each real ω. A typical result calculated by nuerical root searching in the coplex- k plane is shown in Fig.. 4

5 In the sae figure, we copare the coplex band structures for lossy ( γ =., results display as grey-colored lines) and lossless ( γ =, results display as blue-colored lines) etal with electric perittivity, εω = ω ωω+ iγ, where γ is the electron ( ) p /[ ( )] scattering rate. Chosen for an easy coparison with the results in the literature, 7 3 the plasa frequency for the Drude etal is are chosen to be ωp = 6.8 ev and the geoetrical paraeters a = 75 n and r = 5 n. Such paraeters will reproduce the band structure in Ref. 3 outside the light cone (i.e., k > ω c ) in the case of γ =. The / h band structure has a negative slope for a range of k close to the zone boundary. It should be noted that such a band of negative slope always exists when the particles are close enough. The slope of the band will turn to positive when the inter-particle distance becoes larger. When there is no absorption, there is a critical frequency ωc 3.7 ev below which there are two distinct guided ode solutions [ k and k, where Re( < Re( ]. In this regie, the guided odes define a pass-band region because I( = I( =, indicating that the plason excitation is truly guided along the chain without radiation loss. As frequency increases to ω > ωc, we have Re( = Re( and I( = I(. (This is due to the tie reversal syetry.) This region is a band-gap region in which the plason odes are leaky and can radiate to the surroundings. When we have γ, all solutions have I(. (The two solutions are distinct because the tie reversal syetry is broken by absorptive dissipation.) For ω < ω, the propagating plason c 5

6 experience a loss that is doinated by absorption. As frequency increases to ω > ω, both absorption and radiation losses can suppress the plason propagation. Details on the c features of the band structure are available elsewhere 3 and we will not repeat the discussions. The ai of showing Fig. is to show that the ratio between absorption and radiation losses can be adjusted by considering different frequencies and absorption rates. We can, therefore, find out the siilarities and differences between the two kinds of losses by a suitable choice of paraeters. III. LATTICE GREEN S FUNCTION We now consider the spatial decays of plasonic excitations along the D array. The frequency-doain transverse lattice Green s function, G( ω,, n) G( ω, n), for the coupled dipoles satisfies M n ) G( ω, n) n ( ω = δ. (4) n The response dipoles for a given external driving field are thus given by p (, ) ext = G ω n En. G( ω, ) has the usual eaning of the response of the syste n ext ext when one particle is excited by a localized source ( En = δne ) and can be written as a Fourier integral: 8 a π / a ika G( ω, ) = α eig ( ω, e dk, (5) π π / a where α ( ω, eig = [/ a 3 α ( ω) κ( ω, ] is the Green s function in the reciprocal space. 8 It should be noted that G( ω, ) = G( ω, ) because of the irror syetry. While the coplex band structure does not display the coplete wave nature of the 6

7 syste (such as the exact response to the external field), the Green s function keeps all the inforation about the waves in the syste. Therefore, all properties shown in the coplex band structure can be fully explained by investigating G( ω, ) and we can see the actual eaning of the coplex k by coparing with G( ω, ). There are at least two ways to evaluate G( ω, ). We can substitute the analytical forula of α ( ω, into Eq. (5) and evaluate the integral. However, a closed-for solution to eig the integral is not available and nuerical evaluation of the integral ay run into stability probles. 8 Therefore, we choose to calculate, nuerically, the inverse of M by truncating M to a atrix of finite diension ( N N ), which is the sae as considering a finite chain of N particles. In order to preserve the irror syetry, we consider N as an odd nuber. Then, the Green s function is G(, ) ω = ( M ) δn ( M ) n n =, where = N, N +,, N, N with N = ( N )/. Our ethod is siilar to those in Refs. 7 and 8 except the single-site driving electric field is located at the center of the chain so that the irror syetry between the two ends of the chain is kept [see Fig. (a)]. In our calculations, we take N = 3. Here, we first briefly highlight soe features of G( ω, ). When there is no absorption and radiation loss (i.e., γ = and ω < ω ), G( ω, ) does not decay. As shown in Ref. 8, c the results are intuitively obvious and thus the results are not repeated here. However, when the wave experience ainly absorption loss (i.e., γ =. ev and ω < ω ), the Green s function, G( ω, ), has two short-range exponential decays (which doinates c 7

8 for < ) and a long-range power-law decay ( > ) [see Fig. (b)]. Due to the beating between odes of different wave nubers, we can observe oscillations at the crossovers between different decays. In the case of pure radiation loss (i.e., γ = and ω > ω c ), G( ω, ) has only one short-range exponential decay and a long-range powerlaw decay [see Fig. (c)]. In addition to the beating at the crossover points, there are strong oscillations for the whole exponential decay curve in Fig. (c). Such oscillations are associated with the tie reversal syetry and will be explained later in this paper. When radiation loss co-exists with absorption loss (i.e., γ =. ev and ω > ωc ), the result [see Fig. (d)] shows no qualitative difference fro that of Fig. (b) except that the exponential decay lengths are shortened. These interesting features (single/double exponential decays with power-law decay) have been largely ignored (though they indeed exist if we exaine the details) in previous published works. 7 8 In the following, we will understand such features by aking an ansatz for the for of G( ω, ) and the accurate eaning of the coplex Bloch wave vector will eerge accordingly. IV. ANSATZ AND JUSTIFICATIONS We will use an ansatz of the lattice Green s function, G( ω, ) that takes the for, 9 Gans (, ) c z + c z + f ( ) z ω, (6) where c, c, z, z, z are -independent coplex nubers satisfying z z z and f is a bounded even function satisfying li p f ( ) = = for soe real nuber p. The first two ters in Eq. (6) will give short-range exponential decays while the third ter represents the long-range power-law decay. We will justify 8

9 the ansatz in the following, by coparing G( ω, ) and G ans ( ω, ) and their generating functions in the coplex nuber plane. By considering the generating functions, we will get extreely accurate values of the coplex quantities (including c, c, z, z, and z ) without using any function-fitting ethod. To verify that G( ω, ) has to take the for as shown in Eq. (6), we consider the generating function of G( ω, ), defined as = g ( ω, z) G( ω, ) z, (7) where z is a coplex nuber. By ultiplying z to Eq. (4) and suing the equation over the index, one can show that the exact g( ω, z) reads ika g( ω, e ) = α ( ω,, (8) eig for any coplex nuber k by allowing analytic continuation (as in the usual definition of the polylogorith functions). Thus, we have a closed-fro expression for g( ω, z) and it has a one-to-one correspondence to α ( ω,, which is also the Green s function in the eig reciprocal space. To present a concrete picture, we plot g( ω, z) in the coplex- z plane (see Fig. 3) for various frequencies. Fro Fig. 3, we see that the exact generating function, g( ω, z), has 4 poles (at z = z, z, z, and z ) and two branch cuts for each frequency. The poles are the solutions of Eq. (3) and the branch cuts, which show up as straight line segents, are defined by, Arg( z) = ka for < z < and Arg( z) = ka for z >. These branch cuts are the principle branch cuts of the polylogarith functions. ika The advantage of considering g( ω, z) as a function of z ( = e ) instead of α ( ω, as 9 eig

10 a function of k is that all the inforation are copressed within a finite unit circle, z, because of the irror syetry of the MNP chain [i.e., g( ω, z) = g( ω, z ) ]. In Figs. 3(a) and 3(b), we see that all four poles lie on the unit circle when there is no loss, i.e. ω < ω and γ =. As we increase ω (keeping γ = ), z oves towards c z while z oves towards z, and all four poles reain on the unit circle. After the poles eet at ω = ω (not shown), they leave the unit circle separately [see Fig. 3(c) and 3(d)], c keeping z = z < (due to radiation loss). It should be noted that we have z = z (due to tie reversal syetry) when γ =. When γ, the absorption breaks the tie reversal syetry, leading to z z and sooth changes in the positions of the poles as frequency changes [see Figs. 3(e)-3(h)]. All these results can be directly apped into the coplex band structure and since each z can be apped into a wave nuber, k, the poles of the generating functions also give all the inforation required to produce the coplex band structure in Fig.. Let us now consider the corresponding generating function for our ansatz Green s function in Eq. (6), g (, ) (, ) ans ω z Gans ω z. Using Eq. (6), we can write = z z z z gans ( ω, z) = c c f( ) z z + +. (9) z z z z z z z z = It should be noted that, in writing Eq. (9), we have again used the analytic continuation as in the polylogarith functions. Fro Eq. (9), we see that the two short-range exponential decays in Gans ( ω, ) are associated with the four poles located at z = z, z, z, and z, which is consistent with g( ω, z) (as shown in Fig. 3). Here, we consider the exact

11 values of the poles and the corresponding residues of g( ω, z). These quantities can be found accurately as long as we have rough estiates for the poles (for exaple, finding the poles fro Fig. 3 by naked eye). With the estiated (and thus approxiate) values of the poles, we can calculate a siple contour integral R = g( ω, z) dz/πi nuerically around a closed curve, C, that encloses the pole but not crossing the branch cuts. We thus obtain the four residues (denoted by R, R, R 3, and R 4 ) of g( ω, z) accordingly. It is expected that these four residues should be consistent with that of g ( ω, z). By C ans equating the residues of g( ω, z) and gans ( ω, z) (i.e. R = cz, R = cz, R3 = cz, and R = cz ), we can solve for c, c, z, and z. The values of z and z obtained 4 in this way are surprisingly accurate in the sense that they pin the poles of g( ω, z) with a relative error within even when the relative error of our initial estiated values is about.. This verifies that the for of the ansatz is essentially exact. To further ake sure that the coplex quantities, c, c, z, and z, can describe the initial exponential decays of G( ω, ) accurately, we also copare cz and cz with G( ω, ) [see Figs. (b), (c), and (d)]. [We have c = c and z = z in the case without absorption loss and thus cz and cz overlap in Fig. (c).] The results deduced fro the ansatz show an exact agreeent to G( ω, ). We note that in addition to the exponential decay(s), there are oscillations in the agnitude of G( ω, ). We entioned that those features near the crossover points are due to the beatings between different odes. Furtherore, there is an additional

12 oscillatory behavior for the whole range of exponential decay in the case of no absorption because we have cz cz + cz = Re( ), which oscillates in its agnitude, when γ =. To verify these stateents, we subtract both cz and cz fro G( ω, ) and plot the reaining function, Gre( ω, ) G( ω, ) cz cz, (dashed lines) in Fig.. We can see that the oscillatory features are iediately reoved after the subtraction and the reaining function is sooth in its agnitude. This is another evidence to show that the ansatz gives the correct functional for of the Green s function. Finally, we have to check whether Gre( ω, ) is of the for, f ( z ), with z =. The verification of z can be done by aking a Fourier transfor of the noralized function, G ( ω, )/ G ( ω, ). We take a finite list of G ( ω, )/ G ( ω, ) for re re to calculate the Fourier transfor nuerically and find that the Fourier re re transfor has only one single sharp peak located at k k (free-space wave nuber), h ikha which eans we have G ( ω, )/ G ( ω, ) e. This is consistent with fixing re re z ikha = e in our ansatz. Furtherore, if Gre(, ) f( ) z ω =, we will have G (, ) ( ) re ω = f. We can see in the insets of Figs. (b)-(d) that Gre ( ω, ) is a p soothly decreasing function that behaves like / for soe constant p, anifesting as straight lines in the log-log plots (except for < ). Again, this verifies the for of the ansatz. The long-range power-law decay has a wave nuber very close to that of the free-space photons, suggesting that such portion of decaying wave is not confined within the chain and is less related to the guided odes. The power factor, p, is found to be

13 close to.35, which is larger than the power factor for the Green s function of photons in free space. V. DISCUSSION AND CONCLUSION In suary, we have shown sei-analytically that the (transverse) lattice Green s function for a periodic etal nanoparticle chain is of the for G( ω, ) = c z + c z a + f ( ) e ik h, which contains both exponential and power-law decays. This is independent of whether the decay is due to radiation or absorption loss. Our results show that a non-zero iaginary part of the Bloch wave nuber k in the coplex band structure only eans an initial short-range exponential decay in the Green s function for such lossy syste. There exists an additional power-law long-range decay that has no apparent relation to the values of k of the guided ode. The existence of such power-law region suggests that extra care ust be taken when one try to obtain the decay length by fitting the Green s function with exponential function. Our results can be helpful in understanding the wave propagation in lossy open systes. In addition, one interesting and subtle feature is that there are two exponential decay constants in the transverse ode, and it is true for frequencies that can excite guided odes or the leaky odes at the band gap above guided ode frequencies. The existence of two (instead of one) decay constants is due to the breaking of tie reversal syetry by absorptive dissipation. If there is no absorption, the two decay lengths becoe equal, as required by tie reversal syetry. 3

14 ACKNOWLEDGMENTS This work was supported by the Central Allocation Grant fro the Hong Kong RGC through HKUST3/6C. Coputation resources were supported by the Shun Hing Education and Charity Fund. We thank Dr. Dezhuan Han, Dr. Yun Lai, and Prof. Zhaoqing Zhang for coents and useful discussions. 4

15 REFERENCES M. J. O. Strutt, Ann. d. Physik 85, 9 (98); 86, 39 (99); F. Bloch, Zeits. f. Physik 5, 555 (98). L. P. Bouckaert, R. Soluchowski, and E. Wigner, Phys. Rev. 5, 58 (936). 3 For photonic bands, see N. Stefanou, V. Karathanos, and A. Modinos, J. Phys.: Condens. Matter 4, 7389 (99) and T. Suzuki and K. L. Yu, J. Opt. Soc. A. B, 84 (995). 4 For electronic bands, see H. J. Choi and J. Ih, Phys. Rev. B 59, 67 (999) and J. K. Tofohr and O. F. Sankey, Phys. Rev. B 65, 455 (). 5 E. N. Econoou, Green's Functions in Quantu Physics, 3rd ed. (Springer, Berlin, 6). 6 P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenoena, nd ed. (Springer, Heidelberger, 6). 7 W. H. Weber and G. W. Ford, Phys. Rev. B 7, 549 (4). 8 V. A. Markel and A. K. Sarychev, Phys. Rev. B 75, 8546 (7). 9 See, e.g., W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 44, 84 (3). The propagation of surface plason in a linear array of etal nanoparticles has been studied experientally in Ref. and theoretically in Refs. 7, 8, 3, 4, and 5. J.R. Krenn, A. Dereux, J.C. Weeber, E. Bourillot, Y. Lacroute, J.P. Goudonnet, G. Schider, W. Gotschy, A. Leitner, F.R. Aussenegg, and C. Girard, Phys. Rev. Lett. 8, 59 (999); R. de Waele, A. F. Koenderink, and A. Polan, Nano Lett. 7, 4 (7); A. F. Koenderink, R. de Waele, J. C. Prangsa, and A. Polan, Phys. Rev. B 76, 5

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17 FIG. : (Color online) Coplex band structure for a D array of MNPs. Left (right) panel show the real (iaginary) part of the wave nuber. The nubers inside the square boxes indicate different non-degenerate odes when absorption is included. 7

18 FIG. : (Color online) Lattice Green s function G( ω, ). Panel (a) shows a scheatic diagra with a dipole source located at the center of the MNP chain. Other panels show the values of G( ω, ) and the coparisons with the functions aking up the ansatz for the case when energy loss is (b) doinated by absorption ( ω = 3.6 ev and γ =. ev), (c) purely due to radiation ( ω = 3.75 ev and γ = ), and (d) contributed fairly by absorption and radiation ( ω = 3.75 ev and γ =. ev), respectively. The insets show the sae graphs but in log-log scale, highlighting the long-range power law decay. Nubers with square boxes indicate the corresponding plason odes shown in Fig.. 8

19 FIG. 3: (Color online) Contour plot of g( ω, z) / r at different frequencies. The 3 poles at z = z, z, z, and z of the function are labeled. Upper (lower) panels show the results for the lossless (lossy) etal with γ = ( γ =. ev). (a) and (e) ω = 3.6 ev. (b) and (f) ω = 3.7 ev. (c) and (g) ω = 3.75 ev. (d) and (h) ω = 3.8 ev. 9