Scanning confocal microscopy of carbon nanotubes 

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Structural and electronic properties

Single Wall Carbon Nanotubes (SWNT) are hollow cylinders that can be considered as graphene sheets wrapped around themselves. Graphene is an allotrope of carbon consisting in a single layer of atoms organized in an hexagonal pattern. First, some information are given about the properties of this material and then the properties of carbon nanotubes are derived from them.

Exciton in a bulk semiconductor

Let’s consider a bulk semiconductor with a direct gap ǫg . The valence and conduction bands have a parabolic dispersion characterized by the effective masses m∗c and m∗v , and centered in ~k = 0. In the fundamental state, the valence band (below the Fermi level) is full and the conduction band (above the Fermi level) is empty. If one considers that there are no interactions between charge careers (between electron and hole), the first excited state is the one for which an electron is promoted from the valence to the conduction band, and the energy to provide for such an event is equal to ǫg. However, in a simple picture, one may expect the electron and the hole left behind in the valence band – which are of opposite charge – to interact via Coulombic force. This interaction leads to bound states called excitons, with an energy lower than the energy of the gap. Let’s consider a very simple model for the exciton, taking into account only the electron-hole interactions. In the barycentric frame, the problem can be replaced by the one of a fictive particle in a central force motion [98]. This pseudo-particle has an effective mass M = m∗c + m∗h (where the effective mass of the hole is m∗h = −m∗v ). By analogy with the hydrogen atom, the total energy of the exciton (indexed by quantum number α∈ N) is given by : ǫα(−→K) = ǫrenorm g + ~2K2 2M − R∗ α2.

Synthesis of Carbon Nanotubes

Carbon nanotube are scarce in the natural environment, but probably exist in common fires [126]. Their first artificial synthesis was made in the group of Ijima in 1991 [13], using an arc-discharge evaporation method, with graphite electrodes under high temperature and low pressure. This first method, which had previously been established for the production of fullerene had the disadvantage of creating many different carbon allotropes, including amorphous ones. Carbon nanotubes rapidly drew the interest of researchers, as well as industrials, due to their amazing physical properties. This gave birth to a entire field of research, still very active now, on their synthesis. The efforts are taking two directions : first a high purity is sought (this means reducing the presence of amorphous carbon, limiting crystalline defects). Second, a control of the geometrical properties of carbon nanotubes (their diameter, or chiral angle) is the aim of many researcher. The selectivity (meaning to capacity to choose the (n,m) indices of the nanotubes grown) remains limited, and is often based on post-selection. All the synthesis methods have their pros and cons. They are all briefly introduced, though the nanotubes used during this PhD all came from the same commercial sources.

Micellar suspension

Though carbon nanotube were synthesized since 1991, their photoluminescence was observed for the first time at the beginning of the 21st century. The reason is that they were produced as a powder in which the nanotubes formed small bundles (a specific case can be seen in figure 2.14). These aggregates hold by Van der Waals bonds. Though the forces involved are low, their integration over the full nanotubes length leads to very strong binding energies, typically over a few eVμm−1, which means over thermal agitation, even at room temperature. The issue is that bundles lead to interactions between semi-conducting and metallic nanotubes, which in turn lead to a quenching of the fluorescence of the formers [130].
In 2002, O’Connell et al. proposed a way to break these bundle in order to individualize carbon nanotubes [16]. For that, an aqueous dispersion of raw single wall carbon nanotubes was diluted in sodium dodecyl sulfate (SDS) and ultrasonic agitation was performed. Afterwards, they used centrifugation to remove the remaining bundles. The nanotubes left were encapsulated in cylindrical micelles, as depicted in figure 2.15. In these micelles, the hydrophobic part of the SDS molecules is against the tube, and the hydrophilic one is towards the outside, which eases the solubilization.

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Cavity Planar-Concave modes

The Fabry-Perot interferometer is a textbooks classics : if two mirrors face each other, some stationary modes are established in between. These modes depend on the curvature of the mirrors, their reflectivity and the distance between them. A discussion of the longitudinal resonant condition is given, as well as a description of the spatial properties of these modes.

Table of contents :

Introduction
1 Spontaneous Emission Control with a Cavity 
1.1 Light – Matter Coupling
1.1.1 Atom in free-space
1.1.2 Cavity coupling: the Jaynes-Cummings model
1.2 Purcell effect
1.2.1 Purcell factor
1.2.2 High quality factor cavities
1.3 Purcell effect in condensed matter
1.3.1 Purcell factor for a condensed matter emitter
1.3.2 Light confinement in semi-conductor structures
1.3.3 The case of carbon nanotubes
2 Single Carbon Nanotube Properties 
2.1 Structural and electronic properties
2.1.1 Structural properties
2.1.2 Electronic properties
2.2 Optical properties
2.2.1 Single electron model
2.2.2 Excitonic properties
2.3 Samples
2.3.1 Synthesis of Carbon Nanotubes
2.3.2 Micellar suspension
2.3.3 Spin coating
3 A Fiber Fabry-Perot Microcavity 
3.1 Cavity Planar-Concave modes
3.1.1 Resonant condition
3.1.2 Spectrum of an empty cavity
3.2 Cavity mode volume
3.2.1 Manufacturing of fibered mirrors
3.2.2 Effective mode volume
3.3 Fiber – Cavity coupling
3.3.1 Single mode fibers
3.3.2 Beyond single mode fibers
3.4 Finesse and storage time
3.4.1 Losses
3.4.2 Measuring the finesse
4 Scanning confocal microscopy of carbon nanotubes 
4.1 A scanning Confocal Microscope
4.1.1 Principle
4.1.2 Excitation
4.2 Low temperature single nanotube photoluminescence
4.2.1 Single carbon nanotube photoluminescence
4.2.2 Coupling to acoustic phonons
4.3 Spectral diffusion in Carbon Nanotubes
4.3.1 Spectral diffusion of the ZPL
4.3.2 Spectral diffusion of the Phonon Wings
5 A Purcell enhanced single-photon source 
5.1 Coupling a tunable cavity to the confocal microscope
5.1.1 Cavity Setup
5.1.2 Lens-Fiber mounting principle
5.2 Measuring the Purcell Factor
5.2.1 Photon counts method
5.2.2 Time domain method
5.2.3 Experimental Results
5.3 Single-photon source
5.3.1 Statistics of a light source
5.3.2 Second order correlation function
5.3.3 Hanbury-Brown and Twiss setup
6 Efficiency of a cavity coupled nanotube 
6.1 Cavity efficiency including the phonon wings : theoretical approach .
6.1.1 Evolution of the populations
6.1.2 Cavity efficiency
6.2 Experimental derivation of the cavity efficiency
6.2.1 Experimental “reconstructed” spectra
6.2.2 Fit of the experimental data
6.2.3 Evolution with cavity volume
Conclusion
References 
Glossary

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