# Support

## Support Options

• ### Knowledge Base

Find information on common issues.

• ### Ask the Community

Ask questions and find answers from other users.

• ### Wish List

Suggest a new site feature or improvement.

• ### Support Tickets

Check on status of your tickets.

## CNT Bands Challenge Problem

Version 11
by Denis Areshkin
Version 12
by Denis Areshkin

Deletions or items before changed

Additions or items after changed

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:''' 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)]] + 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)]] + + Figure below plots analytical expression for localization length (blue line) along with the simulated localization length, which has been statistically averaged over an ensemble of 750 randomly distorted CNTs (green line). Localization length is measured in numbers of unit cells of CNT. The unit cell is defined as the minimum size unit cell, which interacts only with its nearest neighbors. Red line is the transmission of the ideal (undistorted) CNT in arbitrary units. + + [[Image(Fig_9_1110_Localization_Length_One_Channel_01.png,50%)]]

### Legal

nanoHUB.org, a resource for nanoscience and nanotechnology, is supported by the National Science Foundation and other funding agencies. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.