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In real life most fractures occur at defects. Even such everyday materials as cement can have their strength increased many times by reducing the defect density within them. You'll often see claims for the incredible strength of nanostructures, but the strength is just due to the fact that these structures are free of defects. It's a lot easier to make a ...

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Graphene is also very thin. According to this article the force required to break a sheet of perfect graphene by pulling it apart in such a way that all bonds break at the same time is 42 N per meter. If the width of your tape is 1cm you would need to apply 0.42 N. It is not surprising that you were able to. Even if the sheet would be perfect, you would pull ...

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I guess you are confused by the action on wave functions and operators. 1) The second approach to your first concern is "more correct." Notice that $t(k)$ or $w(k)$ is a $c$-number( or scalar, not an operator). When you write $w(k) \rightarrow w(-k)$, I guess you treat it as a wave function (I am not sure, because if it is a wave function, it should ...

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In any case, for undoped graphene, the Fermi level of electrons and holes are symmetric so the Fermi Energy lies at the Dirac Point, so these two definitions would be equivalent.

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