Tight-binding equations for a semi-infinite graphene sheet
It’s quite straightforward how to solve TB-equations for an infinite graphene sheet (see Castro Neto et al., Rev. Mod. Phys. 2009):
![E_{\mu k}\alpha (k,n) = -t[(1+e^{ika})\beta (k,n)+\beta(k,n-1)]](http://s.wordpress.com/latex.php?latex=%20E_%7B%5Cmu%20k%7D%5Calpha%20%28k%2Cn%29%20%3D%20-t%5B%281%2Be%5E%7Bika%7D%29%5Cbeta%20%28k%2Cn%29%2B%5Cbeta%28k%2Cn-1%29%5D%20&bg=ffffff&fg=000000&s=0 "E_{\mu k}\alpha (k,n) = -t[(1+e^{ika})\beta (k,n)+\beta(k,n-1)]")
,![E_{\mu k}\beta (k,n) = -t[(1+e^{-ika})\alpha (k,n)+\alpha(k,n+1)]](http://s.wordpress.com/latex.php?latex=%20E_%7B%5Cmu%20k%7D%5Cbeta%20%28k%2Cn%29%20%3D%20-t%5B%281%2Be%5E%7B-ika%7D%29%5Calpha%20%28k%2Cn%29%2B%5Calpha%28k%2Cn%2B1%29%5D%20&bg=ffffff&fg=000000&s=0 "E_{\mu k}\beta (k,n) = -t[(1+e^{-ika})\alpha (k,n)+\alpha(k,n+1)]")
.
One should search a solution (for both 
![\exp [ i\mu a n ]](http://s.wordpress.com/latex.php?latex=%5Cexp%20%5B%20i%5Cmu%20a%20n%20%5D%20&bg=ffffff&fg=000000&s=0 "\exp [ i\mu a n ]")
How to find a solution in this case ? It seems that the usual trick with a standing wave instead of a running wave doesn’t work…
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