How to explain difference in C-N and C-B interactions?

[Quantamania]

Fellows,
     I have returned with a new challenge regarding my hexagonal boron nitride/graphene composites.  I and my mentor are trying to treat these systems using first-order perturbation theory, by defining interlayer interactions as separable and distinct from each other.  In the unit cells, we have only B...C and N...C interactions.  We also have intrabond interactions, as a primary source for band degeneracy in graphene and polarized bonds in h-BN layers.  Based on our eigenstate pictures for both materials, we found larger orbitals for boron compared to nitrogen.  We focus on the energy levels near K of the Brillouin zone, where the band gap is formed in our composites and Dirac cones exist for graphene.  Basically, we control how far the apexes in the Dirac cones are from each other in these composites.  This produced one of our initial assumptions about the contributions arising from B...C and N...C interactions.

However, when I analyzed the composites by looking at each instance either of these interactions occur in the unit cells, I was able to reproduce observed band gap size trends when the N...C interaction contributes more to energy level perturbations than the B...C interaction does.  Trying the other way, relying on orbital size alone, fails to reproduce the finding that a NB C staggered bilayer has a larger band gap opening (0.0637 eV for a DZDP 40x40x20 LDA grid) than a BN C staggered bilayer (0.0399 eV with the same parameters).  In the NB C staggered bilayer, the N atom is situated above a C atom in the graphene layer, leaving the B atom situated above a hole in the ring.  The BN C staggered bilayer has the opposite situation.  We also encounter problems for trilayer conformers with regards to how large the band gap is, depending on how many of these interactions appear in the unit cell, when we initially assume a B...C interaction is stronger than a N...C interaction.

I have performed electron density contour cuts through a BN C eclipsed bilayer, provided in the post (whose band gap is a sum of both staggered bilayers' band gaps), observing that the N atom expands outwards more than the B atom.  This means that the probability density at nitrogen atoms expands further out than that at boron atoms.  In our first-order perturbation theory treatment, we assume that each N...C interaction pushes the graphene energy level upwards (destabilizes it), while B...C interactions are attractive and move the energy level to lower energies.  Each interaction is kept at the same distance from each other, and these interactions clearly show exponential dependence on distance between layers.  Adding a new layer for trilayer conformers retains the additivity of these interactions, but also introduces cancellation by dipole-dipole opposition.

As each B...C interaction is essentially an empty orbital interacting with a half-occupied orbital (for graphene there is a degenerate pair of orbitals and we are filling them half-way with electrons to simulate its metallic behavior), and a N...C interaction is a lone pair interacting with a half-occupied orbital, there is a difference between these two interactions in terms of electron-electron interactions.  One is an one-electron interaction (indeed attractive but weak), while the other is a three-electron interaction (repulsive and can be strong).

Here is my question for you: Are there reasons as to why I am getting a larger band gap when I allow N...C interactions, compared to B...C interactions?  This will really help me a lot with my dissertation and can help me answer challenges from my mentor and committee alike.  I wanted to discuss this because of the initial assumptions failing to make sense (how can a larger atomic orbital not induce a stronger energy shift compared to a smaller one, unless there are differences between these orbitals in terms of another property?).

[zh]

Your question is too long.  Anyway, the following basic comments may be helpful for you to understand your problems:
i). If we just think about the substitution for the dopant in graphene, the B doping graphene will act as hole doping; while N doping in graphene, it acts as electron doping.
ii). The Pauling electronegativities of B, C, and N are 2.04, 2.55, and 3.04 respectively. see http://www.webelements.com/nitrogen/electronegativity.html
iii). To understand the effect of B or N doping on the electronic structures of graphene, it is better to first have a clear picture on the characteristic of eigenstates near Fermi level (or the conduction band edge and the valence band edge). If the eigenstates near the Fermi level come from the p orbitals of carbon atom in graphene, the B or N doping may significantly affect the electronic structure of graphene because the p electrons in B or N atom are one less or larger than that in carbon atom.

[Quantamania]

Yes, according to the eigenstate pictures corresponding to K in the bilayers and trilayers, we frequently see isolated graphene orbitals, in those with a significant band gap.  In the valence band, we tend to see single pi orbitals situated near one set of atoms, while in the conduction band, we see the other set of pi orbitals present in the eigenstate.  In those without this band gap, the energy levels are degenerate and give different-looking eigenstates.  We never get mixing of orbitals at K, because the BN layer bands stay far away from the graphene bands around this point, by as much as 2.5 eV relative to the Fermi energy level.  Thus, this is clearly an indication of doping effects at a long range.  In our models, the layers are kept at 3.34 angstroms apart.  The band gap is easily tuned by interlayer distance, due to exponential decay of the pi orbitals away from the origin.

I have already applied electronegativity differences among these elements in my discussions, so I was looking for other kinds of reasons.  The electron-doping for C...N interactions appears to be stronger than the hole-doping that C...B interactions inflict upon the band gap of graphene.  What influences the strength of the doping effect that a heteroatom has on the orbitals of nearby atoms?

[zh]

Rigidly, one p electron due to the N doping will occupy  the conduction band edge (or the the lowest unoccupied molecular orbital(LUMO)) of pure graphene, so the band gap of N-doped graphene turns to be the energy difference between the n+2nd and n+1st eigenvalues of pure graphene where n is the index of eigenvalue for the valence band edge (the highest occupied molecular orbital (HOMO)) or pure graphene. Similarily, one p electron due to the B doping will be removed from the valence band edge (or the HOMO) of pure graphene, so the band gap of B-doped graphene  is still the energy difference between the n-th and the n+1st eigenvalues of the pure graphene. This is just a picture of rigid band. Of course, the B or N doping will change the exact eigenvalues of graphene.