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The first report of high-quality CdSe nanorods demonstrated linear polarization of emission along the rods’ long axes using a stretched polymer film.62 Polarized emission of CdSe nanorods was subsequently studied using single particle fluorescence

spectroscopy, finding photoluminescence polarizations, / , of up to

0.86.160 Atomic force microscopy measurements and fluorescence polarization

*_{Adapted with permission from: B.T. Diroll, T. Dadosh, A. Koschitzky, Y.E. Goldman, C.B. Murray.}

Interpreting the Energy-Dependent Anisotropy of Colloidal Nanorods Using Ensemble and Single-Particle Spectroscopy. J. Phys. Chem. C2013, 117 (45), 23928-23937. Copyright 2013. American Chemical Society.

measurements on single nanorods show that the maximum of emission polarization lies along the nanorod long axis.161 The relationship of emission polarization to the nanorod geometry is typically explained by exciton fine structure. CdSe nanorods form in the hexagonal wurtzite crystal structure of CdSe, with the long-axis of the nanorod growing along the c-axis of the crystal.62 Similar to quantum dots, CdSe nanorods comparable in size to the Bohr exciton radius of CdSe show absorption features arising from quantum confinement but in only two dimensions. In ensembles, size heterogeneity leads to inhomogeneous broadening of the absorption features. For single nanorods, each absorption feature consists of several closely-spaced electronic states accounting for the allowed energies of a photogenerated electron-hole pairs (excitons). The lowest excited state of CdSe nanocrystals, which is responsible for photoluminescence, has a well- studied exciton fine structure. The exciton fine structure of the lowest excited state of CdSe consists of eight states defined numerically by the total angular momentum of the exciton (hole plus electron). They are ±2, ±1L, ±1U, 0L, and 0U, of where the superscripts denote upper (U) and lower (L) states of the same momenta and + and – indicate spin handedness.162 The ±2 and 0L are optically forbidden transitions not expected to contribute significantly to room temperature emission.

Photoluminescence derives from radiative relaxation of the exciton fine structure states to the ground state according to the oscillator strength of the transitions weighted by the excited state populations. For CdSe nanocrystals, room temperature emission is a mixture of recombination from the 0U _{excitonic state, with photons linearly polarized} along the wurtzite c-axis, and the ±1 state, with photons circularly polarized in the ab plane.155,163–166 In CdSe nanorods, photoluminescence polarized along the wurtzite c- axis—the nanorod long axis—dominates the ±1 transitions, whereas in CdSe quantum dots the opposite pattern is true.151,154,165,166 Theory and experiment suggest that both the oscillator strength and the population of the 0U excitonic state explain this phenomenon. Shabaev and Efros show the oscillator strength of the linearly-polarized transitions increases with aspect ratio and that the anisotropic dielectric environment of the nanorod further increases the dominance of linearly-polarized emission.151 Separately, measurements of single CdSe nanorod photoluminescence at low temperature show high

polarization (>0.95) of one feature attributed to 0-state recombination and a shortening of the photoluminescence lifetime at 5 K compared to 80 K.153 This suggests that the lowest excitonic state, with the highest population fraction, is not the forbidden ±2 state, as in quantum dots,162 but is rather an optically-allowed transition of linear polarization. As the exciton fine structure depends on the shape of the lattice at the nanometer length scale, linear photoluminescence of CdSe nanorods may be considered a confinement effect similar to the change in the band-gap energy with size.

Nanorod absorption polarization receives less attention and remains less thoroughly understood. Although the emission profile and polarization remains fixed regardless of the energy of excitation, the polarization of nanorod absorption is sensitive to the excitation energy. Two-photon excitation measurements demonstrate a polarized lowest-

energy absorption in CdSe nanorods167 and wavelength-dependent polarization

measurements of single nanowires show the role of quantum-confined states in absorption polarization.168 In this report, we follow the work of Sitt et al., Hadar et al., and Tice et al., who analyzed colloidal semiconductor nanorods using the photoselection method.161,169,170 There are two potential sources of polarization in CdSe nanorod absorption: classical dielectric effects and quantum transition dipoles. The absorption cross-section for a transition from a state i to a state f is

| | · | | ₁ (3.1)

in which · is the transition dipole, εmedium is the dielectric constant of the surroundings, and ε is the dielectric constant of the absorber.171,172 _{The factor | · |} reflects directionality in the state-to-state transition and α is the depolarization factor, measuring differential attenuation of incident electric fields caused by a dielectric in

which . (See below)

The interplay of quantum and classical sources of polarization is much-debated with regard to CdSe nanowires,173,174 but measurements of quantum-confined nanowires show distinctive polarized excitonic features like those in Figure 3.4.168 A description of dielectric absorption anisotropy is given below, but applying dielectric models to low-

energy transitions is difficult because the full-spectrum dielectric function of nanoscale semiconductors is unknown and dielectric models are least relevant to low-energy absorptions, where excitations are well-separated and easily distinguishable.169,172 In the fitting procedure below, we focus on the low- energy excitations in which transition dipoles dictate anisotropy.

3.3 Ensemble Measurements of Optical Anisotropy