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There's been much recent interest in 2D materials since they can form monolayer-thick films. Since their crystal structure is periodic along the in-plane directions, the electronic band structure along these directions is quite well understood and can be formulated using approaches used for conventional bulk semiconductors, such as Si.

How can we think of the out-of-plane electronic band structure (which is not periodic) for 2D materials?

I need help bridging the techniques used in solid-state physics for bulk crystalline materials to the calculation of band structure along a non-periodic direction. My initial thought is that the notion of band structure along the out-of-plane direction is ill-conceived.

The reason why I'm interested in the concept of out-of-plane band structure, is that devices have been proposed and fabricated that have electron transport (tunneling) from one 2D layer to another, and the concept of band structure is quite useful in applying conventional techniques to calculate current, rates, etc. [For example, this 2014 paper discusses tunneling between different layers of 2D materials, but uses the in-plane band structure for the out-of-plane direction.]

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You're right, the usual language of band theory doesn't apply in the out-of-plane direction. The system really becomes a quantum well in that direction, so you will have a discrete spectrum of energy levels and there won't be any dispersion with $k_z$ ($dE/dk_z = 0$) if $z$ is out of the plane.

The relevant tunneling processes in that paper are basically occurring between a double quantum well formed by the two 2D layers with the barrier in between. The depth of each well depends on the band structure; where the electrons sit in the 2D Brillouin zone (what momentum or equivalently wavevector $\vec{k}$ they have in the plane) and the 2D band structure will dictate their energies. In the case of semiconductors, they're assuming that only the lowest valleys of the conduction bands (highest peaks of valence band) will have appreciable electron/hole density, and by tuning the those to the same level by gating, then tunneling occurs. There's no dependence on $z$ momentum because (except for tunneling processes) the electrons are localized in the 2D planes and there just isn't any $z$ momentum.

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  • $\begingroup$ Thank you for your answer—it's quite helpful. I would add that there should be $z$-momentum just as there is in a quantum well; however, there will only be specific allowed values for $k_z$. (Momentum is typically not conserved along the tunneling direction because of an external electric field.) Do you think it is reasonable to use the free-electron mass along the $z$-direction? $\endgroup$ – JTT Jun 24 '16 at 19:50
  • $\begingroup$ Short answer is yes, because you're just dealing with electron tunneling along that direction. Long answer: Effective mass is the inverse of the band curvature, so in the limit of a truly 2D material the lack of dispersion with $k_z$ means that the effective mass in that direction is technically diverging to infinity. That means it takes an infinite amount of energy to start it moving: it's localized to be a standing wave in the planar quantum well. So since effective mass is a band theory concept and band theory doesn't really apply in the $z$ direction, it's not physical to apply it here. $\endgroup$ – psio Jun 25 '16 at 1:30

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