NANO LETTERS
Classification and Analysis of Excitations in a Coupled Finite Nanoparticle Cylinder-Shell Lattice
2008 Vol. 8, No. 9 2944-2948
Jared M. Maxson and Slava V. Rotkin* Physics Department, Lehigh UniVersity, 16 Memorial East DriVe, Bethlehem, PennsylVania 18015, and Center for AdVanced Materials and Nanotechnology, 5 East Packer AVenue, Bethlehem, PennsylVania 18015 Received June 22, 2008; Revised Manuscript Received July 15, 2008
ABSTRACT We present a general model for computing an optical response function of a finite shell lattice of semiconducting or metallic nanoparticles. Within a second quantization formalism, a cylindrical shell of induced, coupled dipoles is considered in the presence of an external electric field. Numerical analysis of the eigenmodes and quantum mechanical response function allow us to identify resonator effects due to constructive interferometric interaction of the light to the dipole lattice. Adjusting the wavelength of the external electric field, a coherent resonance excitation is possible for a fixed parameter of the cylinder radius.
The use of an induced dipole lattice to characterize the optical properties of crystals was employed by Mahan and Obermair,1 and now provides the theoretical basis of modern studies of Frenkel excitons,2 both in bulk and at surfaces. In applications of the Mahan and Obermair model, the retarded interaction of atomic sites entirely represented by point dipoles is considered.3,4 The use of a dipole lattice to describe an intrinsically more complicated system was also employed by Purcell and Pennypacker,5 further developed by Draine.6 Their method, often termed the coupled dipole method (CDM) or discrete dipole approximation, discretized a homogeneous light scatterer into cubical electrical “domains”, represented entirely by induced, fully coupled electric point dipoles. The polarizability of each dipole is then determined by a modified Clausius-Mossotti relation.7 We propose that the self-assembly of optically active (polarizable) nanoparticles into regular shell-like geometries provides another opportunity for the use of the coupled dipole concept. Rather than using atoms in crystal lattice, and rather than than discretizing a single scatterer into subunits, we analyze a preexisting nanoscale lattice of distant nanoparticles which are still close enough to justify nonretarded interaction. Within our theoretical model, a shell lattice of semiconductor or metallic nanoparticles is substituted with an idealized regular coupled dipole lattice. A small parameter which allows us to keep only the dipole term in our Hamiltonian is the same as in the classical multipole expansion theory.8 The total dipole moment of the shell lattice should dominate the far-field electromagnetic response, thus our method * Corresponding author. 10.1021/nl8018017 CCC: $40.75 Published on Web 08/13/2008
2008 American Chemical Society
allows for a theoretical nanophotonics and metamaterials application theory. Furthermore, the analysis of shell (hollow) systems has practical relevance in that many nanoelements self-assemble into shell structures. Talapin et al.9 suggest that the process of self-assembly of centrosymmetric nanoparticle lattices is dominated by the nanoparticle dipole moment. We find notable production of such lattices by various chemical and self-assembly techniques: rings of nanorods10 and cylindrical shell lattices of gold nanoparticles,11 as well as spherical shells of macroion cages,12 for example. Having been inspired by the success of such synthetic efforts, we develop here the theoretical formalism for a system represented as a shell lattice of coupled dipoles, with the dipole moment written in the second quantization. We compute the optical response of such a system which can be used in analysis of the optical characteristics of self-assembled lattices of nanoelements. Furthermore, shell structures are more sensitive to changes in geometrical parameters than bulk media. For example, we will determine below special optical resonances of the shells that depend on the shell size/ geometry. With this general model formulated, the calculation and analysis of the quantum mechanical response function of a cylindrical-shell lattice of interacting dipoles in the presence of an external electric field is presented. For a given polarization of the incident light, we will determine the cylindrical radii which produce maximal coupling to the external field of the light.
We approximate a shell structure with a uniform lattice of nanoparticles, where possible extension of such an idealized model is discussed at the end of the paper. We assume the lattice constant a is much smaller than the wavelength of the incident light, Λ, to provide a coherent nonretarded excitation. We are interested in modeling optically active modes of the shell lattice. Thus, the dipole elementary excitations of a single nanoelement of the lattice should be considered. We adopt the simplest form of the analogous CDM, in that we substitute a single point dipole for each nanoparticle. In general, a weak coupling to nonradiative (nondipole like) elementary excitations is possible. We argue that for a practical interparticle distance such terms and such coupling must be neglected in the multipole expansion.2 We also assume a single, uncoupled nanoelement of the lattice has an isotropic static polarizability R0, as well as a single resonant excitation frequency ω0. Neither of the last two assumptions is critical for our theory, but the present results and derivation are substantially more transparent in this approximation. The Lagrangian operator of the system in the Heisenberg representation is as follows: 1 L) 2
∑
βγ γ -1 βγ p†β i (R (ω)δijδ + Vij )pj -
i,j
∑
β p†β i Ext,i
(1)
i
where b Ext,i is the external electric field at the site i, b pi is the annihilation operator of the dipole excitation at the site i, Vβγ ij is the dipole propagator, or Green’s function of the dipole-dipole interaction, which, for i not equal to j, is given by Vβγ ij )
1 βγ (δ - 3(r^βijr^γij)) r3ij
(2)
where in the above equations i and j are the indices of the lattice site, ˆr is the unit vector along b, r the dipole separation distance vector, and β and γ define Cartesian components. We note that the single nanoparticle polarizability can be written as R(ω) ) Roω2o/(ω2o - ω2), which is an exact expression for the leading term in the series expansion of a general polarizability near ωo, the resonance frequency of the elementary excitation. We define the reduced conjugate momentum π ) ms ) (m/e)∂tp ) -imωp/e through the effective mass obtained from the sum rule for the oscillator strengths: Ro ) mω2o/e2. Here, s is the velocity operator, and e is the elementary charge. The significance of the terms in the Lagrangian may be seen in bringing them to the canonical form. We substitute the expression for π and R(ω) in the Lagrangian: Lo ) 1 ) 2
1 2
∑p ∑ †β i
i
∑[
j
[ (
R-1 o -
β †β -π†β i πi + pi
i
]
)
ω2 γ + 1 δijδβγ + Vβγ ij pj 2 ωo
∑(
βγ βγ R-1 o δijδ + Vij
j
pγj
)
]
(3)
The first term in the right-hand side (rhs) of eq 3 is the kinetic energy of the field; the second term is the potential energy, constituted by the dipole self-energy and the dipole-dipole coupling energy. Canonical variable substitution gives us the real-space Hamiltonian, where the sign of the first term changes: Nano Lett., Vol. 8, No. 9, 2008
Ho )
1 2
∑ [π
†β β †β i πi + pi
i
∑ (R
-1 βγ βγ o δijδ + Vij
i,j
)pγj
]
(4)
To determine the eigenvectors of polarization, pλ, we must bring the Hamiltonian to diagonal form. As the dipole-dipole interaction is the only nondiagonal term, we may define our eigenset entirely in terms of the dipole operator eigenequation: ^
Vpλ ) Vλ · pλ
(5)
where Vλ and pλ are the eigen-value and eigen-vector, corresponding to the solutions of the eigen-equation above. Thus, we can define the reduced conjugate coordinate as b qλ
) b pλ/√R-1 o +Vλ that would bring the Hamiltonian to the canonical form: Ho )
1 2
∑ [πb πb
† q†λb qλ λ λ+b
]
(6)
λ
The eigenfrequency of each mode can be calculated from the eigen-value of the dipole-dipole interaction. This may be derived directly from eq 3, from which we obtain ω2λ ) ω2o(1 + RoVλ)
(7)
If the symmetry of the lattice is high enough, an analytic transformation can be applied to diagonalize the Hamiltonian (eq 4). Specifically, if a lattice is translationally invariant in the generalized coordinate x, we may write 〈κ|x1〉 〈x1|Ho|x2〉 〈x2|k 〉 ) 〈κ|Ho|k 〉 δκk
(8)
where 〈κ|x〉 ) is the standard rotation matrix from the coordinate x space to the κ momentum space. Using eigenmodes of the lattice excitation, we may write the quantum mechanical response function of any dipole lattice system in the presence of an external field b E (cf. ref 13): (1/N)eiκx
χ(ω) ) 2
e2 V
∑ (ω
|Mλ|2Γ
2 2 λ - ω) + Γ
λ
(9)
where V is the normalization volume, and we define the absorption matrix element as |Mλ|2 ) |〈0|Eˆ · p |λ〉|2 ) |Eˆ · p λ|2 f
f
(10)
where |0〉 is the ground state, and λ is the index of the eigenvector, containing all quantum numbers required to describe the mode |λ〉. Γ is the phenomenological broadening of the peak, small enough to not influence the results. The above formalism is valid for any system of fully coupled dipoles, but we now turn to the analysis of the cylindrical shell geometry. In this paper, we analyze an achiral shell lattice of finite length, such that it is a rectangular square lattice scrolled into a cylinder, to form a succession of identical rings. For this, we define a standard Cartesian coordinate system, such that the z direction is along the axis of the cylinder, with the x-y bisecting the cylinder length. The arbitrarity of the placement of the x and y directions is removed upon diagonalization, in that the resulting modes exhibit dipole distributions oscillating radially or tangentially (as well as along z), rather than in the x or y direction. We 2945
Figure 1. Representation of the geometry of the system as well as the polarization of the incident light, denoted by b E and b k. a is the interparticle distance and R is the cylinder radius.
analyze a plane wave external electric field polarized along the x axis and propagating across the cylinder axis z as shown in Figure 1. In this paper, we only consider lattices of finite size in which the edge effects of the cylinder truncation are not negligible. As such, it is theoretically possible to perform an analytic transformation to partially diagonalize the dipole-dipole interaction about the angular coordinate θ, bringing the matrix to angular momentum space according to eq 8. However, as we have no translational invariance about the z coordinate, the functional form of the V operator involving both θ and z makes such a transformation difficult. Thus, we turn to numerical methods. In using numerical means of diagonalization, we loose direct angular and linear momentum data for each mode, but gain the ability to quickly form the complete response function. We may partially reclaim momentum data from the analysis of the dipole distribution of each mode, as will be shown below. To obtain the response function (eq 9), we first determine the numerical eigen-solution of eq 5 for the shell geometry described above. We perform diagonalization in one step, flattening the fourth rank tensor, Vβγ ij , by placing the polarization indices within the lattice indices. We seek to demonstrate resonant behavior between the external electric field and the lattice excitation, which we define as high coherence (both spatial and temporal) between the field and only one or few eigen-modes for a given wavelength of light. Thus, calculating the total response function and the partial response function of single modes permits quantitative analysis of the resonances. For numerical calculation, we must define the model parameters such as R0 and ωo, the static polarizability and resonant frequency of the nanoelement of the shell, a the shell lattice constant, R and L, shell sizes. We couple the shell eigen-modes with a photon of frequency ω and the wavelength Λ ) 2πc/ω. We will use below unitless variables for the frequency of the incident light ω/ω0 measured in units of eigen-frequency. Similarly, the unitless shell size may be given by 2R/Λ or L/Λ, depending on the polarization of the incident light. Though R0 is material dependent, it may be considered in our calculations as an adjustable parameter, where we specify R0 ) 0.01a3. The sought resonances appear at a fixed ratio between the cylindrical shell radius and the frequency of the single particle. 2946
Figure 2. Response function of the cylindrical coupled lattice for 30 dipoles per ring and 11 rings vs the unitless excitation frequency ω/ω0 and inverse unitless wavelength qj ) 2R/Λ. Line broadening is Γ ) 0.01ω0. The contributing modes may be broken down into “families” of similar qj dependence (see Figure 3 for their partial response functions). The mode wavelength is shown in the insets on the scale of the cylinder cross section. The first maxima of each of the active families in this region are labeled.
In Figure 2, we plot the response function for the cylindrical shell with 30 dipoles per ring and 11 rings long. The vertical axis is the unitless wavevector of the incident light, jq ) 2R/Λ, that is, the ratio of the diameter of the cylinder to the light wavelength, and the horizontal axis is the unitless excitation frequency ω/ω0, measured in units of the eigenfrequency of a single dipole. We stress that even though Λ and ω are not independent variables, the shell radius and ω0 are. The lowest resonances appear in the uniform electric field, jq ) 0, which corresponds to a cylinder of insignificant radius as compared with the wavelength of light. This assures that, for fully symmetric modes, when all dipoles are in phase and oriented along the z direction (there is some perturbance at the ends due to finiteness of the system), the excitation field is also synchronous at all lattice sites. The frequency of the excitable eigen-modes however varies because of substantial dipole-dipole interaction. The remaining resonance regions involve modes of higher geometrical complexity in their distribution of dipoles, thus, cataloging them based on their electric field is difficult. However, we note that these active modes may be broken down based on their jq dependence into what we will refer to as mode “families”. We specify that the modes in each family have identical response function dependence on jq, up to a multiplicative constant. In Figure 3, the family dependence some of the active modes in this region is demonstrated. In each panel, the matrix elements of only the most active modes are plotted. The remaining modes have oscillator strength at least 10 times less than those plotted or are zero. Beyond the first maxima, the amplitude of the oscillation of the partial Nano Lett., Vol. 8, No. 9, 2008
Figure 4. Axial z components of the dipole moment, pz, vs the lattice site index, iθ (on each cylindrical ring), and the axial site coordinate z (separate, rising curves), for four different modes (separate graphs) of the second mode family. The orbital wavelength is the same for all modes in a family.
Figure 3. Logarithmic plot of |M|, the matrix elements of the first three qj mode families vs the inverse unitless excitation wavelength qj ) 2R/Λ. A logarithmic scale is used to demonstrate similar resonant behavior of the mode, but hides the strong decay of response function magnitude at increased qj .
response function for each mode decreases significantly; however, as this happens, new active families emerge and dominate the response, such as family 4 in the jq ) 2.5 regions. In this area, families 1, 2, and 3 have oscillator strength at least 10 times less than that of those in family 4. This pattern continues as jq increases until all modes and symmetry families have been exhausted. This decay and emergence of modes and mode families with changing jq accounts for both the periodic “spotting” in Figure 2, which are contributions of single or few modes at scattered frequencies, and the seemingly continuous resonance regions, which are contributions from many modes and many families at similar eigen-frequencies. Modes constituting a family are determined by identical angular symmetry of the z component of dipole moment. Plotted in Figure 4 are the z components of the site dipoles on each ring as a function of angular and axial coordinates, for different modes constituting one mode family. Unlike the distributions of dipoles along the z axis which differ between modes in a family and are nonperiodic, Figure 4 shows perfect (discrete) sinusoidal angular behavior of equal wavelength for all modes in a family. This result suggests not only that excitations may be resolved into components of angular momenta (via the Fourier transform or otherwise) Nano Lett., Vol. 8, No. 9, 2008
as proposed above, but also that the familial behavior of the modes for this polarization is defined by eigen-modes having equal excitation angular momentum, but different linear momentum. Changing the system size does not alter this general behavior; the inclusion of more dipoles per ring merely permits the existence of more symmetry families (more allowed angular momentum values), and the inclusion of more rings causes an increase in the number of modes per family. Because of both the spatial and frequency coherence of the electric field and the dipole distribution, only 11 modes contribute significantly throughout the entire plot region out of the total 990 modes present in the sum of the response function plotted in Figure 2, which given the inclusion of both spatial and temporal factors in the response function definition, satisfies our definition of resonant behavior. In conclusion, we have presented a quantum mechanical model for use in the optical characterization of nanoelement shell lattices. We have described the transformation of the second quantized Hamiltonian of the shell lattice to its eigenspace, and in doing so determined the shifting of the single particle resonance frequency ω0 due to lattice coupling. We propose that the model may be used to predict behavior of such shell lattices of optically excitable nanoelements. As an example of such a nanoelement system, we consider the optical response function of a cylinder lattice-shell of dipoles, in which we treat the cylindrical radius a tunable parameter. We demonstrate resonant coupling between light polarized perpendicularly to the cylindrical axis and the lattice excitation in noting the limited number of eigen-modes contributing significantly to the strong peaks of the response function. Our model enables the determination of cylindrical radii that provide maximal coupling to the external field. In analyzing the response function, we find identical wavelength dependence in groups of eigen-modes (families), which we correlate to eigen-mode groups of the same angular momentum, and thereby the same angular symmetry. Interesting properties of such nanoelement shells will be determined by lattice geometry. The resolution of the above eigen-modes into groups of angular momenta, given the imperfect translational invariance of the finite cylinder, 2947
suggests that successful symmetry analysis (either analytic, as in the Fourier transformation of the Hamiltonian, or numerical as in this case) may be performed on lattices of only partial symmetry to determine or explain novel optical properties. We stress that our model is readily generalized for a more realistic shell geometry with nonideal lattice structure. The lattice defects as well as the nonuniform screening of each of the particles by its randomized environment can be included via a nonuniform dipole propagator, which will replace eq 2 and change the off-diagonal matrix elements of the Hamiltonian. Anisotropic and nonuniform polarizabilities of single particles will modify the diagonal matrix elements. Then, the eigen-solutions would be different from the ideal case; however, for a small randomness in the matrix, they will resemble the symmetry of the modes of the perfect lattice shell. Acknowledgment. This work was partially supported by DoD-ARL (Grant W911NF-07-2-0064) under Lehigh-Army Research Laboratory Cooperative Agreement, by the National Science Foundation (Lehigh REU program), by the American Chemical Society (ACS PRF 46870-G10). References (1) (a) Mahan, G. D. Van der Waals Forces in Solids. J. Chem. Phys. 1965, 43 (5), 1569–1574. (b) Mahan, G. D.; Obermair, G. Polaritons at surfaces. Phys. ReV. 1969, 183 (3), 834–831.
2948
(2) (a) Rotkin, V. V.; Suris, R. A. Plasmon-Frenkel-exciton in a clustered solid. Phys. Lett. A 1998, 245 (3), 292–296. (b) Rotkin, V. V.; Suris, R. A. Vacuum-ultraviolet plasma Frenkel excitons: elementary excitations of polymerized fullerene. Phys. Solid State 1998, 40 (5), 839– 841. (3) Jenkins, J. K.; Mukamel, S. Quantum electrodynamics of molecular nanostructures. J. Chem. Phys. 1993, 98 (9), 7046–7058. (4) (a) Philpott, M. R.; Sherman, P. G. Excitons and polaritons in monomolecular layers. Phys. ReV. B 1975, 12 (12), 5381–5394. (b) Kempa, K.; Ruppin, R.; Pendry, J. B. Electromagnetic response of a point dipole crystal. Phys. ReV. B 2005, 72, 205103. (5) Purcell, E. M.; Pennypacker, C. R. Scattering and Absorption of Light by Nonspherical Dielectric Grains. Astrophys. J. 1973, 186, 705–714. (6) Draine, B. T. The discrete-dipole approximation and its application to interstellar graphite grains. Astrophys. J. 1988, 333, 848–872. (7) Yurkin, M. A.; Hoekstra, A. G. The discrete dipole approximation: An overview and recent developments. J. Quant. Spectrosc. Radiat. Transfer. 2007, 106, 558–589. (8) Jackson, J. D. Classical Electrodynamics; Wiley: New York, 1998; 3rd ed. (9) Talapin, D.; Shevchenko, E.; Murray, C.; Titov, A.; Kral, P. Dipoledipole interactions in nanoparticle superlattices. Nano Lett. 2007, 7 (5), (10) Khanal, B. P.; Zubarev, E. R. Rings of nanorods. Angew. Chem., Int. Ed. 2007, 46 (13), 2195–2198. (11) Zubarev, E.; Xu, J.; Sayyad, A.; Gibson, J. Amphiphilicity-driven organization of nanoparticles into discrete assemblies. JACS 2006, 128 (47), 15098–15099. (12) Li, D.; Zhang, J.; Landskron, K.; Liu, T. Spontaneous self-assembly of metal-organic cationic nanocages to form monodisperse hollow vesicles in dilute solutions. JACS 2008, 130 (13), 4226–4227. (13) Herrendo¨rfer, D.; Patterson, C. H. Dipole waves in semiconductors: the dielectic function and plasma oscillations of silicon. J. Phys. Chem. Solids 1997, 58 (2), 207–220.
NL8018017
Nano Lett., Vol. 8, No. 9, 2008
