CNT Bands Challenge Problem
 Version 11
 by (unknown)
 Version 12
 by (unknown)
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1  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.  

2  
3  '''Given:'''  
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5  In a first nearest neighbor piorbital tightbinding approximation let us assume that random distortion is described by random shifts $\backslash Delta\_\{ii\}$ of onsite Hamiltonian matrix elements $h\_\{ii\}$. The dispersion $\backslash sigma$ of onsite shifts is defined in its usual way:  
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7  $\backslash sigma^\{2\}=<\backslash Delta\_\{ii\}^\{2\}><\backslash Delta\_\{ii\}>^\{2\}$,  
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9  where angular brackets denote averaging over all atoms ''i''.  
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11  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.  
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13  '''Find:'''  
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15  For the given CNT indexes ($n\_1$, $n\_2$) and dispersion $\backslash sigma$ find the localization length as function of electron energy.  
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20    The article contains detailed and selfcontained explanation of quantum transmission calculations. As an example providing insight on the properties of Greens functions and contact selfenergies, analytical expression for localization length in randomly distorted CNT is derived (Section VIII, Eqs.(4143)). Download: [[File(RealLifeProblem_CNT.pdf)]] 
+  The article contains detailed and selfcontained explanation of quantum transmission calculations. As an example providing insight on the properties of Greens functions and contact selfenergies, analytical expression for localization length in randomly distorted CNT is derived (Section VIII, Eqs.(4143)). Download: [[File(RealLifeProblem_CNT.pdf)]]

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22  +  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.


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24  +  [[Image(Fig_9_1110_Localization_Length_One_Channel_01.png,50%)]] 