F.2.1 The Phonon GREEN's Function

The density-density correlation function of the nuclei is reduced in the following to a quantity of more practical interest, namely the phonon GREEN's function within the harmonic approximation. One can expand the ionic charge density up to first-order in the lattice displacement $ {\bf u_\alpha(L\kappa)}$ with respect to the equilibrium positions of ions (see (F.25)) [205]

\begin{displaymath}\begin{array}{l}\displaystyle \rho_\mathrm{n}({\bf r},t) \ = ...
...\bf L}-{\bf\kappa}) u_{\alpha}({\bf L\kappa},t) \ , \end{array}\end{displaymath} (F.38)

where $ \alpha$ denotes the Cartesian components. This expansion reduces the density-density correlation function (F.34) to

\begin{displaymath}\begin{array}{l}\displaystyle D_{\alpha\beta}({\bf r},t;{\bf ...
...\rho_{\kappa'}({\bf r'}- {\bf L'}-{\bf\kappa'}) \ , \end{array}\end{displaymath} (F.39)

where the phonon GREEN's function in real space is

\begin{displaymath}\begin{array}{l}\displaystyle D_{\alpha\beta}({\bf L\kappa},t...
...},t)\ u_{\beta}({\bf L'\kappa'},t')\ \} \rangle \ . \end{array}\end{displaymath} (F.40)

Owing to the lattice periodicity of the ionic charge densities, the spatial FOURIER transformation of (F.39) takes the form

\begin{displaymath}\begin{array}{l}\displaystyle D_{\alpha\beta}({\bf G}+{\bf q}...
...} (G_{\beta}'+q_{\beta}) \ e^{i{\bf G'\kappa'}} \ . \end{array}\end{displaymath} (F.41)

The FOURIER expansion of the lattice displacement can be written as

\begin{displaymath}\begin{array}{l}\displaystyle u_{\alpha}({\bf L \kappa},t) \ ...
...+{\bf\kappa})}u_{\alpha{\bf \kappa}}({\bf q},t) \ , \end{array}\end{displaymath} (F.42)

where $ M_{\bf\kappa}$ are the mass of the atoms and $ N$ is the number of atoms in the unit cell. By means of (F.42), the FOURIER transformation of (F.40) is given by

\begin{displaymath}\begin{array}{l}\displaystyle D_{\alpha{\bf\kappa},\beta{\bf\...
...f q'},t') \ \} \rangle\delta_{{\bf q},{\bf q'}} \ . \end{array}\end{displaymath} (F.43)

By diagonalizing the dynamical matrix [283], one obtains the eigen-modes $ e_{\alpha\kappa \lambda}({\bf q})$ and eigen-frequencies $ \omega_{\lambda,{\bf q}}$ of the lattice vibrations. These eigen-vectors can be used to expand the FOURIER components of the displacement in terms of phonon operators

\begin{displaymath}\begin{array}{l}\displaystyle \hat{u}_{\alpha\kappa}({\bf q},...
...}}(t) + b^\dagger_{\lambda,-{\bf q}}(t) \right) \ , \end{array}\end{displaymath} (F.44)

where these operators have the time dependence

\begin{displaymath}\begin{array}{l}\displaystyle b_{\lambda,{\bf q}}(t) \ = \ b_{\lambda,{\bf q}} e^{-i\omega_{\lambda,{\bf q}}t} \ . \end{array}\end{displaymath} (F.45)

This eigen-vector expansion allows one to factorize (F.43) for each phonon mode according to

\begin{displaymath}\begin{array}{l}\displaystyle D_{\alpha{\bf\kappa},\beta{\bf\...
...kappa'}({\bf q};t,t') D_{\lambda}({\bf q};t,t') \ . \end{array}\end{displaymath} (F.46)

into a weight factor

\begin{displaymath}\begin{array}{l}\displaystyle d_{\alpha{\bf\kappa},\beta{\bf\...
...pa'}\lambda}({\bf q})}{2\omega_{\lambda,{\bf {q}}}} \end{array}\end{displaymath} (F.47)

and the phonon GREEN's function (Appendix D.2)

\begin{displaymath}\begin{array}{l}\displaystyle D_{\lambda}({\bf q};t,t') \ = \...
...hat{A}^\dagger_{\lambda,{\bf q}}(t') \} \rangle \ , \end{array}\end{displaymath} (F.48)

where $ \hat{A}_{\lambda,{\bf q}}(t)=b_{\lambda,{\bf q}}(t)+b^\dagger_{\lambda,{-\bf q}}(t)$, $ b$, and $ b^\dagger$ are the annihilation and creation operators for Bosons. This factorization allows one to evaluate the coupling for any combination of phonon branch indices.

M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors