Hollow Shells of Dipoles: A Group Theoretical Approach
Christopher Devulder, Slava V. Rotkin1
1Department
of Physics, Lehigh University, Bethlehem, PA
INTRODUCTION
RESULTS
METHOD
We use Group Theory to solve for the dipole interaction part of the Hamiltonian of
various types of one (chain), two (ring) and three (cylinder) dimensional shells of
dipoles. The fabrication of such structures has been recently achieved for various
plasmonic devices, such as gold nanoparticles surrounding carbon nanotubes (1).
These systems present various levels of rotational and translational symmetry that
can be put in correspondence with symmetry groups and their matrix
representations. This mapping of our physical system onto mathematical groups
enables us to use several powerful results and theorems from linear algebra to
diagonalize the interaction potential tensor that appears in the Hamiltonian. The
resulting eigenmodes of oscillation can thus be obtained analytically with their
corresponding energies.
(1) Bing Li, Lingyu Li, Bingbing Wang and Christopher Y. Li. Alternating patterns on single-walled carbon
nanotubes. Nature Nanotechnology 10.1038/NNANO.2009.9 (2009).
Group Theory
• A group is a mathematical set that contains one or more elements
satisfying certain properties. These elements can be mathematical objects
of various nature: functions, matrices, real numbers etc.
• The group is equipped with a well defined operation that acts on two
group elements and yields a third one, also contained in the group.
• The group and its operation have to satisfy certain properties
(associativity, existence of an identity element, of inverses, etc.)
• e.g. (ℝ,+), (ℂ,x), the set of all invertible 2 by 2 matrices, the set of all
permutations of a list containing 6 elements, or the set of rotational
symmetries of a pentagon.
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BACKGROUND
e
Plasmons
g1
g4
• Plasmons are density waves
of valence electrons, formally
defined as quanta of plasma
oscillations.
• Polaritons result from the the
coupling of a photon to such a
local charge density
fluctuation.
• The eigenvalues thus obtained can be utilized to plot the response function of
our system. In addition, we are able to determine the energy corresponding to
each specific eigenmode of oscillation, which we are able to obtain by our
analytical approach.
g3
g2
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• We can establish a correspondence between the dipole sites that make up
our lattice and its matching symmetry group:
• Top (left): plots of the electric field of 3 different modes (4 dipoles on a ring).
• Top (right): dispersion relations of a chain made of 30 dipoles and its
corresponding density of state displayed on top.
• Below: plot of the response function vs. frequency of a ring with 10 dipoles.
The red curve corresponds to a single mode.
• They are generated at optical frequencies and may propagate very rapidly
under certain circumstances. They can theoretically encode a lot more
information than what is possible for conventional electronics, by behaving like
waveguides.
Hamiltonian
• They can also occur in nanoparticles, where they are treated as point dipoles in
various models.
• However, plasmons tend to dissipate after only a few millimeters, making them
too short-lived to serve as a basis for computer chips, which are a few centimeters
across.
• The key is to use materials with a low
refractive index, ideally negative, such that
the incoming EM energy is reflected parallel
to the surface of the material and transmitted
along its length for as far as possible. This is
when specifically designed nanostructures
come into play.
• The Heisenberg operator of the interaction V for a cylinder of of N rings
each containing M dipoles can be written as:
• This tensor can be diagonalized by performing a unitary transformation of
the type U-1VU to display the eigenenergies of the system on its diagonal.
• This is accomplished by means of a projection operator, whose
mathematical structure is entirely given by group theory and the symmetry
groups corresponding to the geometry of our system.
Acknowledgments

Sherman Fairchild Fellowship & Lehigh University REU Program
CONCLUSIONS
In this work, we have developed a general theoretical model to calculate the
optical response of lattices of polarizable nanoparticles and quantum dots, by
treating the excitation of the single element as a dipole written in the second
quantization. A central focus of the model then becomes the dipole-dipole
interaction tensor, as the Hamiltonian and the interaction tensor are diagonal in
the same representation. This concept is applied to a simple linear lattice, in
which the eigensystem of the lattice is determined analytically, and an analytical
dispersion relationship is determined. Currently we are focusing on extending the
model to a cylindrical shell of dipoles.