336
L. D. Negro et al.
9.2.3 Plasmonic Chains: Collective Excitations and Energy Gaps
The advantage of the substitutional method is that relevant information on the nature
of the diffraction spectra of aperiodic sequences can be obtained from the knowledge of the substitution matrix, whose elements indicate the number of times a
given letter appears in the components of the substitution rules (irrespective of the
order in which it occurs) [13]. This follows from the Bombieri–Taylor theorem,
which unveils a fundamental connection between the arithmetical nature of substitutions and the presence/absence of Bragg peaks in the corresponding Fourier spectra
[44–48]. According to the Bombieri–Taylor theorem [49, 50], if the spectrum of the
substitution matrix S contains a so-called Pisot–Vijayvaraghavan (PV) number as an
eigenvalue, then the sequence is quasiperiodic, 3 otherwise it is not. 4 A very important
question in aperiodic systems research concerns the relation between their structural
properties/topological order and the energy spectra of their elementary excitations
and eigenmodes. Solving this difficult problem is of direct interest to aperiodic optics
and plasmonics, as it will enable the formulation of powerful structural-property relations for predictive device modeling and engineering.
A fundamental result in this direction in known as the gap-labeling theorem.
This theorem relates the positions of the diffraction Bragg peaks of substitutional
sequences with the locations of the gaps in the energy spectra of the elementary excitations supported by the structures (e.g., plasmon modes) [44, 45, 51]. In general,
a tight-binding analysis of the energy spectra or the density of states of aperiodic
structures obtained by Pisot-type substitutions shows that both the position and the
width of the gaps can be “labeled” by the singularities of the Fourier transform
associated to the aperiodic sequence of scattering potentials (optical or electronic)
[44, 51]. This approach, first introduced for the 1D Schrödinger equation [52], has
also been applied to the optical wave equation in quasiperiodic and almost-periodic
structures [21, 51]. Recently, Dal Negro et al. [53] developed an efficient transfer matrix approach, valid within the dipole approximation, for the calculation of
the resonant eigenfrequencies, oscillation eigenvectors and the integrated density of
states (IDS) of chains of metallic nanoparticles (i.e., dipolar chains) with Fibonacci,
Thue–Morse and Rudin–Shapiro aperiodic modulation [32, 53, 54].
Aperiodically modulated plasmonic chains are constructed by letting d A ∼ A and
d B ∼ B in the corresponding substitution rule, where d A = 25 nm and d B = 30 nm
are two minimum interparticle separations. With this identification, we can readily
map symbolic aperiodic sequences into chains of nanoparticles with deterministic
aperiodic order, as sketched in Fig. 9.3 for a Fibonacci chain of Ag nanoparticles.
This system can be modeled by considering the dynamical equation for an arbitrary
Hertzian dipole in the chain [32, 53, 55], and recasting it into a matrix that connects
3 This means that its spectrum can be expressed as a finite sum of weighted Dirac ε-functions,
corresponding to Bragg peaks that are indexed by integer numbers.
4 A PV number is a positive algebraic number larger than one and such that all of its conjugate
elements (i.e., the other solutions of its defining algebraic equation) have absolute value less than
one. For instance, the golden mean, satisfying the algebraic equation x 2 −x −1 = 0 is a PV number.
L. D. Negro et al.
9.2.3 Plasmonic Chains: Collective Excitations and Energy Gaps
The advantage of the substitutional method is that relevant information on the nature
of the diffraction spectra of aperiodic sequences can be obtained from the knowledge of the substitution matrix, whose elements indicate the number of times a
given letter appears in the components of the substitution rules (irrespective of the
order in which it occurs) [13]. This follows from the Bombieri–Taylor theorem,
which unveils a fundamental connection between the arithmetical nature of substitutions and the presence/absence of Bragg peaks in the corresponding Fourier spectra
[44–48]. According to the Bombieri–Taylor theorem [49, 50], if the spectrum of the
substitution matrix S contains a so-called Pisot–Vijayvaraghavan (PV) number as an
eigenvalue, then the sequence is quasiperiodic, 3 otherwise it is not. 4 A very important
question in aperiodic systems research concerns the relation between their structural
properties/topological order and the energy spectra of their elementary excitations
and eigenmodes. Solving this difficult problem is of direct interest to aperiodic optics
and plasmonics, as it will enable the formulation of powerful structural-property relations for predictive device modeling and engineering.
A fundamental result in this direction in known as the gap-labeling theorem.
This theorem relates the positions of the diffraction Bragg peaks of substitutional
sequences with the locations of the gaps in the energy spectra of the elementary excitations supported by the structures (e.g., plasmon modes) [44, 45, 51]. In general,
a tight-binding analysis of the energy spectra or the density of states of aperiodic
structures obtained by Pisot-type substitutions shows that both the position and the
width of the gaps can be “labeled” by the singularities of the Fourier transform
associated to the aperiodic sequence of scattering potentials (optical or electronic)
[44, 51]. This approach, first introduced for the 1D Schrödinger equation [52], has
also been applied to the optical wave equation in quasiperiodic and almost-periodic
structures [21, 51]. Recently, Dal Negro et al. [53] developed an efficient transfer matrix approach, valid within the dipole approximation, for the calculation of
the resonant eigenfrequencies, oscillation eigenvectors and the integrated density of
states (IDS) of chains of metallic nanoparticles (i.e., dipolar chains) with Fibonacci,
Thue–Morse and Rudin–Shapiro aperiodic modulation [32, 53, 54].
Aperiodically modulated plasmonic chains are constructed by letting d A ∼ A and
d B ∼ B in the corresponding substitution rule, where d A = 25 nm and d B = 30 nm
are two minimum interparticle separations. With this identification, we can readily
map symbolic aperiodic sequences into chains of nanoparticles with deterministic
aperiodic order, as sketched in Fig. 9.3 for a Fibonacci chain of Ag nanoparticles.
This system can be modeled by considering the dynamical equation for an arbitrary
Hertzian dipole in the chain [32, 53, 55], and recasting it into a matrix that connects
3 This means that its spectrum can be expressed as a finite sum of weighted Dirac ε-functions,
corresponding to Bragg peaks that are indexed by integer numbers.
4 A PV number is a positive algebraic number larger than one and such that all of its conjugate
elements (i.e., the other solutions of its defining algebraic equation) have absolute value less than
one. For instance, the golden mean, satisfying the algebraic equation x 2 −x −1 = 0 is a PV number.