## CNT Bands Challenge Problem

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1 2 3 4 It is important to have a quantitative model describing how an interaction of the CNT with its environment (e.g. supporting substrate, other nanotubes, polymer matrix, etc.) influences CNT ability to conduct current. One possible mathematical formulation of this physical problem can be stated as follows. '''Given:''' In a first nearest neighbor pi-orbital tight-binding approximation let us assume that random distortion is described by random shifts $\Delta_\left\{ii\right\}$ of on-site Hamiltonian matrix elements $h_\left\{ii\right\}$. The dispersion $\sigma$ of on-site shifts is defined in its usual way: $\sigma^\left\{2\right\}=<\Delta_\left\{ii\right\}^\left\{2\right\}>-<\Delta_\left\{ii\right\}>^\left\{2\right\}$, where angular brackets denote averaging over all atoms ''i''. For each given energy the localization length of a randomly distorted CNT is defined as CNT length, for which the logarithm of the ratio of ideal transmission to the transmission in the CNT subjected to distortion equals 2. '''Find:''' - Given CNT indexes ($n_1$, $n_2$) and dispersion $\sigma$ find the localization length as function of electron energy. + For the given CNT indexes ($n_1$, $n_2$) and dispersion $\sigma$ find the localization length as function of electron energy. ---- The article contains detailed and self-contained explanation of quantum transmission calculations. As an example providing insight on the properties of Greens functions and contact self-energies, analytical expression for localization length in randomly distorted CNT is derived (Section VIII, Eqs.(41-43)). Download: [[File(Real-Life-Problem_CNT.pdf)]]