semiconductor nanocrystal based saturable absorbers for optical switching applications
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Semiconductor nanocrystal based saturable absorbers for optical switching applications James E. Raynolds School of NanoSciences and NanoEngineering University at Albany, State University of New York, Albany NY 12203 Michael LoCascio Evident Technologies, Inc., 216 River Street – Suite 200, Troy, NY 12180
Abstract This document presents experimental and theoretical results of an effort to develop semiconductor nanocrystal doped thin films for optical switching applications. Films doped with high quality PbS and PbSe nanocrystals have been fabricated. Measurements of the absorption spectra are compared with theoretical predictions. A general theoretical framework for treating the optical properties of multi-layer dielectric structures containing thin nanocrystal doped layers is used. A simple model (the particlein-a-sphere) is used as a starting point, however, more sophisticated models (such as those based on k.p theory, or fits to experiment) can easily be incorporated. A prototypical micron-scale Fabry-Perot optical cavity has been constructed and is discussed. Theoretical predictions for the switching behavior of such a structure, upon introduction of the semiconductor nanocrystal material into the optical cavity, are presented. A full-length version of this paper is available at Evident Technologies’ web site.1
Theoretical approach Due to the space limitations of this paper, we focus only on the theoretical approach. We begin by discussing the simple particle-in-a-sphere model. The model consists in making the approximation that the optically excited electron and hole are noninteracting and are confined by a potential which is constant inside the quantum dot and is infinite outside. The model also assumes parabolic bands. Both of these assumptions are known to be inaccurate.2 Our purpose here is simply to use this model as input to the general theoretical framework describing a non-linear optical device. The dielectric susceptibility of a single quantum dot in the particle-in-a-sphere model has the form: const 2l + 1 χ (ω ) = − (1 − 2 f ) + c.c , ∑ ωV bl hω − E nl (a) + iγ
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as discussed in Ref. 3. In the above equation, V is the sphere volume, γ is a phenomenological constant describing homogeneous broadening, a is the sphere radius, E nl (a ) is the exciton transition energy given by: h 2 κ nl E nl (a ) = E g + 2mr a
2
,
( m r is the electron-hole reduced mass, E g is the bulk band gap, and κ nl is the n’th root of the l’th order spherical Bessel function) and f is the conduction band filling fraction which accounts for saturation effects. The filling fraction f obeys the following rate equation:
df f Sσ =− + (1− 2 f ), τ hωD(ω) dt where S is the field intensity, σ is the optical absorption cross section, D(ω ) is the frequency dependent density of (exciton) states, and τ is the lifetime. The following non-linear relation between the field strengths in the matrix material and the quantum dot must be solved self-consistently:4 3ε h Ed = Eh , 2ε h + ε d ( Ed ) where E d and E h
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