Theory of Optical and Ultrafast Quantum Kinetic Properties of Carbon Nanotubes

This work aims to theoretically model and examine the linear and nonlinear optical properties of single-wall carbon nanotubes (SWCNT). Because of their extraordinary mechanical, optical and electrical properties these systems have attracted a lot of scientific interest. In recent years several time-domain spectroscopic studies have been carried out, but a complete (theoretical) understanding is lacking and there are still several controversial results in ultrafast spectroscopy. To build future (opto-)electronic devices, however, a sound and fundamental understanding of ultrafast and non-linear properties of these nanoscale systems is necessary. This is the main focus of this work. The for CNT novel approach combines density matrix theory with the tight-binding method. It allowes to calculate the linear as well as the nonlinear response to optical pulses taking into account many-particle interactions. First of all, the Bloch equations for CNTs on a screened Hartree-Fock level have been derived. The carbon nanotube Bloch equations (CNBEs) contain excitons as elementary optical excitations for weak excitation and verify the importance of Coulomb effects in CNTs. The Hartree-Fock contributions lead to strong band renormalizations and exciton binding energies both in the order of magnitude of the band gap. Second, the optical response of various CNTs of different diameter in the quasi-equilibrium regime was calculated. As a result of the excitation-induced changes of the Hartree-Fock contributions and the plasma screening, strong optical nonlinearities, including gain for high excitation were found. The excitonic resonance experiences a bleaching and red shift. Both effects gradually increase with the strength of the optical excitation. At sufficiently high carrier densities optical Gain is observed. Such gain calculations are a prerequisite for future CNT lasers. The temporal response of CNTs to ultrashort pulses was modeled taking into account the time dependence of the carrier occupations. For non-resonant excitation below the excitonic resonance the total conduction band carrier density adiabatically follows the optical pulse. For these excitation conditions the pump-probe spectra for several CNTs have been obtained. Again, a bleaching and a strong red shift of the resonance (optical Stark effect) is found. To describe ultrafast carrier relaxation a mean-field theory is not sufficient. Hence the screened Hartree-Fock theory was extended to include carrier-carrier scattering. This results in the thermalization of non-equilibrium carrier populations. To accurately describe carrier dynamics on very short timescales below 100 fs a non-Markovian theory is necessary. Therefore, a correlation expansion was performed to account for memory effects that act on short timescales in the femtosecond regime. The intraband thermalization of carriers was described in a one-band and a two-band model. The calculations in the one-band model predict an ultrafast thermalization of the charge carriers on a ∼ 100 fs timescale. An effect of the non-Markovian dynamics are the oscillations of the electron plasma on the same timescale. The main characteristics of the relaxation dynamics are independent of chirality of the CNTs, but depend on their diameter (slower relaxation with increasing diameter, as the Coulomb interaction is weaker in larger diameter CNTs). Further influencing factors are the external dielectric screening (slower relaxation with increased screening, as scattering is less effective) and the total carrier density (faster relaxation with increasing excitation, as more scattering partners are present). The two-band calculations support the intraband carrier relaxation on a ∼ 100 fs timescale. The advantage of the model is the improved description of the carrier excitation. Furthermore, the excitation induced dephasing of the polarization can be described. On resonant excitation at the exciton the polarization dephases on a picosecond timescale (dephasing is faster for higher carrier densities). Publications • E. Malić, M. Hirtschulz, F. Milde, M. Richter, J. Maultzsch, S. Reich, and A. Knorr ”Coulomb effects in single-walled carbon nanotubes” Physica Status Solidi (b) 245, 2155 (2008) • M. Hirtschulz, F. Milde, E. Malić, C. Thomsen, S. Reich, and A. Knorr ”Theory of ultrafast intraband relaxation in carbon nanotubes” Physica Status Solidi (b) 245, 2164 (2008) • E. Malić, M. Hirtschulz, F. Milde, Y. Wu, J. Maultzsch, T.Heinz, A. Knorr, and S. Reich ”Theory of Rayleigh scattering from metallic carbon nanotubes” Physical Review B 77, 045432 (2008) • M. Hirtschulz, F. Milde, E. Malić, S. Butscher, C. Thomsen, S. Reich, and A. Knorr ”Carbon nanotube Bloch equations: a many-body approach to nonlinear and ultrafast optical properties” Physical Review B 77, 035403 (2008) • S. Butscher, F. Milde, M. Hirtschulz, E. Malić, and A. Knorr ”Hot Electron Relaxation and Phonon Dynamics in Graphene” Applied Physics Letters 91, 203103 (2007) • E. Malić, M. Hirtschulz, F. Milde, Y. Wu, J. Maultzsch, T. Heinz, A. Knorr, and S. Reich ” Theoretical approach to Rayleigh and absorption spectra of semiconducting carbon nanotubes” Physica Status Solidi (b) 244, 4236 (2007) • E. Malić, M. Hirtschulz, F. Milde, A. Knorr, and S. Reich ”Analytical approach to optical absorption in carbon nanotubes” Physical Review B 74, 195431 (2006) Conference Contributions • F. Milde, S. Butscher, M. Hirtschulz, E. Malić, and A. Knorr ”Theory of nonthermal phonon and electron relaxation dynamics in graphene” NOEKS08, Klink/Müritz, Germany, May 2008 • E. Malić, M. Hirtschulz, F. Milde, M. Richter, Janina Maultzsch, S. Reich, and A. Knorr ”Coulomb effects in carbon nanotubes” International Winterschool (IWEPNM08), Kirchberg, Austria, March 2008 • M. Hirtschulz, F. Milde, E. Malić, S. Butscher, C. Thomsen, S. Reich, and A. Knorr ”Carrier-carrier interaction in SWCNTs: Theory of non-equilibrium and ultrafast nonlinear optical properties” International Winterschool (IWEPNM08), Kirchberg, Austria, March 2008 • M. Hirtschulz, F. Milde, E. Malić, S. Butscher, C. Thomsen, S. Reich, and A. Knorr ”Influence of the Coulomb interaction on linear and nonlinear optical properties of SWCNTs” DPG, Berlin, Germany, February 2008 • F. Milde, S. Butscher, M. Hirtschulz, E. Malić, and A. Knorr ”Phonon induced carrier relaxation and hot phonon dephasing dynamics in graphene” DPG, Berlin, Germany, February 2008