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We present theoretical investigations on linear optical spectra and the temporal non-equilibrium dynamics of single-walled carbon nanotubes. In particular we discuss the importance of the surrounding medium. Our approach is based on the density matrix theory, which in combination with tight-binding wave functions allows the calculation of linear and nonlinear optical properties of arbitrary nanotubes. Its strength lies in the straightforward way to describe non-equilibrium properties, such as Coulomb driven ultrafast intrasubband carrier relaxation. We find a strong influence of the surrounding dielectric medium on the position of the peaks, the excitonic binding energy, and the carrier relaxation time. Furthermore, simple scaling laws describing the dependence on the dielectric background constant e bg are shown.

We present microscopic calculations of the excitonic absorption coefficient for single-wall carbon nanotubes. Our approach combines the density matrix formalism including the Coulomb and electron-light interaction with the tight-binding approximation. It allows the investigation of excitons in carbon nanotubes of arbitrary chiral over a wide range of energy. We take all intra- and intersubband contributions into account and study the behavior of the Coulomb matrix elements for all relevant processes. The Coulomb interaction is found to decrease strongly with increasing momentum transfer perpendicular to the nanotube axis. Furthermore, we show that the band gap renormalization arising from the electron-electron interaction leads to a partial lifting of degeneracy in the spectra of zig-zag tubes.

Ultrafast intraband carrier relaxation in carbon nanotubes can be described by Carbon Nanotube Bloch Equations (CNBE) treating the Coulomb interaction beyond the Hartree-Fock level to include carrier-carrier scattering. We obtain equations of motion for carrier densities and two-particle carrier correlations that are solved numerically. The complex non-Markovian relaxation dynamics of hot electrons is calculated for various carbon nanotubes of different diameter and chirality. All calculated nanotubes show an ultrafast relaxation to equilibrium within approximately 100 femtoseconds. The observed relaxation is only weakly dependent on chirality, but depends on the diameter of the nanotube. With increasing diameter of the CNT the relaxation is slowed down, because of the weaker Coulomb interaction in larger diameter CNT. Additionally, we find oscillations of the electron plasma that are due to memory effects in the electron equilibration.

A microscopic calculation of the Rayleigh scattering cross section and the absorption coefficient for arbitrary single-walled carbon nanotubes is presented. The approach is based on the density matrix formalism combined with the tight-binding band structure. Both Rayleigh and absorption spectra show a characteristic intensity ratio behavior, which can be explained by an interplay of the optical matrix element and the joint density of states.

We illustrate the potential of the density matrix theory for investigation of optical properties of arbitrary single-walled carbon nanotubes (CNTs). We have performed microscopic calculations of excitonic absorption spectra for CNTs of different chiral angles and diameters. The obtained results are in good agreement with experiments, in particular the excitonic binding energies match well both experiments and ab initio calculations. Furthermore, we show the strength of our approach by presenting calculations of the ultrafast Coulomb driven non-equilibrium dynamics in CNTs. We find excitation induced dephasing on the picosecond time scale depending on the excitation strength. (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We evaluate a density matrix theory for the description of ultrafast relaxation processes in low-dimensional carbon nanostructures. The theory is based on Bloch equations describing the temporal dynamics of charge carrier population and transition probabilities. In combination with tight-binding wavefunctions, the approach allows the microscopic calculation of linear and nonlinear optical properties of graphene and carbon nanotubes with arbitrary chirality. This way, we have access to time- and momentum-resolved relaxation dynamics of non-equilibrium charge carriers. We study absorption spectra in graphene and carbon nanotubes illustrating the importance of excitonic effects in these structures including the formation of exciton-phonon induced side-bands in carbon nanotubes. Furthermore, we illustrate the relaxation of optically excited charge carriers toward equilibrium via electron-phonon and electron-electron scattering. We observe an ultrafast thermalization of excited carriers within the first hundred femtoseconds followed by a cooling of the electronic system on the picosecond time scale. Moreover, we investigate phonon-induced intersubband relaxation between the two energetically lowest transitions in nanotubes leading to a better understanding of photoluminescence excitation (PLE) experiments.

A method for studying spatially and temporally resolved many-body dynamics of charge carriers in carbon nanotubes is presented. We derive coherent, spatially inhomogeneous Bloch equations for charge-carrier dynamics in an optically excited carbon nanotube. By solving the equations of motion numerically under spatially inhomogeneous excitation conditions, we also demonstrate a striking difference in carrier drift velocity for excitation at the exciton resonance and above the band edge.

We present theoretical investigations of the ultrafast relaxation dynamics and the excitation-induced dephasing of excitonic polarization in single-walled carbon nanotubes. Our microscopic approach is based on the density-matrix theory combined with tight-binding wave functions allowing the calculation of tubes with arbitrary diameter and chiral angle. For excitations to the single-particle continuum, the initial nonequilibrium carrier distribution is thermalized within approximately 100 fs while the polarization adiabatically follows the excitation pulse. For excitations at the excitonic resonance, we find a dephasing of the excitonic polarization on a picosecond time scale depending on the excitation strength. With increasing field intensities, the dephasing time linearly decreases, which is in agreement with recent experiments.

We present a microscopic calculation of the Rayleigh (elastic light) scattering spectra for single-walled carbon nanotubes. Combining density-matrix formalism and the tight-binding model, we obtain results for nanotubes of arbitrary chiral index. In agreement with experiment, the calculated Rayleigh spectra of metallic nanotubes show a double-peaked structure resulting from the trigonal warping effect, with the low-energy feature exhibiting greater intensity. We explain the intensity ratios by the wave-vector dependence of the optical matrix elements. Also in accord with experiment, the Rayleigh line shape of each peak is asymmetric, with an enhanced cross section for lower photon energies. This behavior arises from contributions of the nonresonant optical susceptibility to the Rayleigh scattering process.

Carbon nanotube Bloch equations are proposed to analyze the many-body electron dynamics for optical interband transitions in carbon nanotubes. As a first approach, the Bloch equations for microscopic transitions and occupations are discussed within the screened Hartree-Fock approximation. The potential of the Bloch equation approach is illustrated for linear and nonlinear optical spectra and ultrafast electron dynamics in carbon nanotubes.

We present an analytical approach to the optical absorption in arbitrary carbon nanotubes. We show the chirality dependence of the absorption coefficient to be the result from both the JDOS and the optical matrix element.

We derive an analytic expression for the optical matrix elements of carbon nanotubes and calculate their optical absorption spectra within the tight-binding approximation. For zigzag nanotubes we present an analytical result for the absorption coefficient. Metallic nanotubes have an optical band gap regardless of their chiral angle. The optical absorption intensity increases by up to 10% from zigzag to armchair nanotubes. This trend is in agreement with experiment, although its magnitude is underestimated. We predict a strong absorption peak in the high-energy spectra of certain zigzag nanotubes that can be used to identify the tubes experimentally.

Using density-matrix theory, we microscopically calculate the relaxation dynamics of photoexcited electrons in graphene. Electron-phonon coupling leads to an initially ultrafast energy dissipation and to a nonthermal phonon occupation of the highest optical phonon modes. We also calculate the temporal evolution of the electronic temperature and find good agreement with recent experimental work.