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Transcript

Demonstration of Optical Resonances in a Cylinder Shell Lattice of Quantum Dots Jared Maxson Slava Rotkin Lehigh University Department of Physics J. Maxson, S. V. Rotkin, APS March Meeting 2009 Motivation: Geometries of Self-Assembly • Order on the nanoscale: A pretty A pretty picture picture • Regular geometry is a principle virtue in the self-assembly. • Questions: – For optically polarizable elements, what is the effect of lattice geometry on the system response? – For plane wave excitation, is a resonant interaction between the optical field and exciton system possible? • Our particular geometry: J. Maxson, S. V. Rotkin, APS March Meeting 2009 Modeling the Hollow Nanoelement Cylinder • We consider a cylinder-shell lattice of polarizable nanoelements (nanoparticles, quantum dots) – Modeled as a lattice of induced, fully coupled point dipoles (the excitons) written in the second quantization, with a single excitation dipole moment and a single transition frequency. The lattice constant is assumed to be much smaller than the wavelength of the light to provide coherent excitation. • Subsequent Hamiltonian is analogous to the classical Coupled Dipole Method (Purcell and Pennypacker) for the calculation of the optical response of dielectric with arbitrary geometry: H p2 E1 p1 E2 two elements: E2 ( x1 ) Vˆx1x2 p2 ( x2 ) E0 ( x1 ) Dipole-Dipole Interaction Propogator J. Maxson, S. V. Rotkin, APS March Meeting 2009 Model (Continued) • The fully coupled Hamiltonian of the system is: – Greek indices represent Cartesian components, and Roman indices represent lattice sites. ** – Note: Hamiltonian and interaction propogator are diagonal in the same representation • For simplicity, we assume: – Isotropic polarizability, – Non-retarded interaction propogator: However, such assumptions are not critical to the model. – Could also include: multiple transition frequencies, anisotropic polarizability, fully retarded dipole interaction and quantized field operator (gives rise to polariton effects). • We treat the cylindrical radius as a parameter to identify geometry dependent interaction. J. Maxson, S. V. Rotkin, APS March Meeting 2009 Methods of Calculation • As H is diagonal in the same space as V, diagonalization of the interaction matrix yields the eigenset of polarization: • A translationally invariant Hamiltonian may be diagonalized by rotation to Fourier space: – However, we consider cylinders of finite length, and thus the lattice is not fully translationally invariant. We therefore diagonalize numerically, loosing linear and angular momentum data for each mode. • Diagonalization with numerical methods is straightforward but not directly informative. • How do we classify excitation modes, and determine possible resonant interaction? J. Maxson, S. V. Rotkin, APS March Meeting 2009 Response Function and Resonance • We introduce the quantum mechanical response function for such a system to quantify resonant behavior. – Resonant behavior: defined by high spatial and temporal coherence of the incident light and mode(s) of the lattice. J. Maxson, S. V. Rotkin, APS March Meeting 2009 Response Function Analysis • Matrix elements of modes with non-negligible oscillator strength at a specific peak are plotted together (right). • We note that at each peak the contributing modes have identical radius/wavelength dependence, and are small in number (about 10 in 990). • By our definition, the transfer of significant oscillator strength to a limited number of modes is evidence of resonant interaction—high spatial and temporal coherence of mode and incident wave. • As the field is polarized along the axis of the cylinder, we analyze the angular dependence of pz for those modes with identical radial dependence: • Near-perfect sinusoidal behavior indicates sharp values of mode angular momenta. • Mode “families” are modes of identical angular momentum, but differing linear momentum. J. Maxson, S. V. Rotkin, APS March Meeting 2009 Conclusion and Summary For hollow cylindrical case: • Formation of the quantum mechanical response function permits determination of regions of cylinder size and light frequency that yield maximal system response. • Sharp values of angular momentum in cylinder excitations are preserved, though translational symmetry is broken. – Permits quantitative excitation classification. • For axial polarization, modes with equal angular momenta are grouped together at absorption peaks, contributing to the resonant behavior of high mode selectivity. In general: • A second quantization model is proposed for the determination of the optical response of a strongly coupled polarizable nanoelement lattice (for any geometry). • Methods of determining the eigen-excitations are proposed for both infinite lattices of high symmetry, and finite lattices of reduced symmetry. • What’s next? Less general, more specific? J. Maxson, S. V. Rotkin, APS March Meeting 2009