Christopher Devulder, Slava V. Rotkin 1 Christopher Devulder, Slava V. Rotkin 1 1 Department of...
1
Christopher Devulder, Slava V. Rotkin Christopher Devulder, Slava V. Rotkin 1 1 1 Department of Physics, Lehigh University, Bethlehem, PA INTRODUCTION INTRODUCTION Hollow Shells of Dipoles: A Group Theoretical Approach 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). Plasmons Plasmons BACKGROUND BACKGROUND RESULTS RESULTS CONCLUSIONS CONCLUSIONS Acknowledgments Acknowledgments Sherman Fairchild Fellowship & Lehigh University REU Program METHOD METHOD Group Theory 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. e g 1 g 2 g 3 g 4 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 • 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. • 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. • 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. Hamiltonian Hamiltonian • 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 -1 VU 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. • We can establish a correspondence between the dipole sites that make up our lattice and its matching symmetry group: 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. • 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. • 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.
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Transcript of Christopher Devulder, Slava V. Rotkin 1 Christopher Devulder, Slava V. Rotkin 1 1 Department of...
Christopher Devulder, Slava V. RotkinChristopher Devulder, Slava V. Rotkin11 1Department of Physics, Lehigh University, Bethlehem, PA
INTRODUCTIONINTRODUCTION
Hollow Shells of Dipoles: A Group Theoretical Approach
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).
PlasmonsPlasmons
BACKGROUNDBACKGROUND
RESULTSRESULTS
CONCLUSIONSCONCLUSIONS
AcknowledgmentsAcknowledgments Sherman Fairchild Fellowship & Lehigh University REU Program
METHODMETHODGroup TheoryGroup 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.
e
g1
g2g3
g4
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10000
01000
00100
00010
00100
00010
00001
10000
01000
00010
00001
10000
01000
00100
01000
00100
00010
00001
10000
• 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.
• 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.
• 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.
HamiltonianHamiltonian• 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.
• We can establish a correspondence between the dipole sites that make up our lattice and its matching symmetry group:
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.
• 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.
• 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.
