Gap modification of atomically thin boron nitride by phonon mediated interactions
© Hague; licensee Springer. 2012
Received: 17 January 2012
Accepted: 23 April 2012
Published: 14 June 2012
A theory is presented for the modification of bandgaps in atomically thin boron nitride (BN) by attractive interactions mediated through phonons in a polarizable substrate, or in the BN plane. Gap equations are solved, and gap enhancements are found to range up to 70% for dimensionless electron-phonon coupling λ =1, indicating that a proportion of the measured BN bandgap may have a phonon origin.
The need for bandgaps in graphene on electronvolt scales has led to a number of proposals, such as the use of bilayer graphene , creation of nanoribbons , and manipulation through substrates [3, 4]. Recently, it has become possible to manipulate atomically thin layers of boron nitride (BN) and other materials with structure similar to graphene . This may lead to a complimentary method of manipulating bandgaps to make digital transistors.
In low dimensional materials, strong effective electron-electron interactions can be induced via an interaction between electrons confined to a plane and phonons in a polarizable neighboring layer . The theory has shown that similar interactions account for the transport properties of graphene on polarizable substrates  and that sandwiching graphene between polarisable superstrates and gap opening substrates can cause gap enhancement . This paper examines similar gap changes in atomically thin BN due to interactions mediated through substrates.
For simplicity, the Holstein electron-phonon interaction was used, , which qualitatively captures the physics. There may be quantitative changes to the results for longer range Fröhlich interactions and from modulation of the electron-phonon interaction due to incommensurability of the substrate, which was estimated at around ±8% of the average value .
Results and discussion
The local approximation used here is a good starting point because the modulated potential Δ is large, and electrons are well localized. Off-diagonal terms do not feature in the lowest order perturbation theory for the Holstein model since the interaction is site diagonal. Z n is the quasi-particle weight and is the gap function. For bosonic quantities, and for fermions, . T is the temperature and n and s are integers.
where the full gap is . The density of states for a tight binding hexagonal lattice in the absence of a gap, D (ε ), has the form given in reference . The equations may be solved self-consistently by performing a truncated sum on Matsubara frequencies.
A theory for the modification of BN band-gaps by interaction with phonons was presented here. It is of interest to make a comparison between the bandgaps of bulk h-BN, nanotubes, monolayer h-BN, and the theory presented here. Measured bandgaps of bulk h-BN are of between 5.8 eV  and 5.971 eV , indicating that interaction between layers increases the bandgap, consistent with the theory here. The bulk gap is also higher than that for nanotubes (5 eV) . On the other hand, Song et al.  claim that the gap is reduced as BN thickness increases. The above discussion is presented with a warning that the theory requires that hopping between the substrate and the BN monolayer is small. Interlayer hopping will affect the bandwidth and bandgap, and the direct Coulomb interaction with strongly ionic substrates could also affect the band structure if the charge density at the surface of the substrate varies dramatically.
It is also of interest to estimate the magnitude of the bandgap modification due to electron-phonon interaction in isolated monolayers of BN. Ab initio calculations have attempted to quantify the magnitude of the interaction between electrons and acoustic phonons for small momentum excitations . Extrapolating the interaction and taking a mean-field average (assuming mean momentum magnitude of 4π /9a ), the electron-phonon coupling can be estimated as , taking E1=3.66 eV from reference , amu, a=2.5Å. The mean energy of longitudinal acoustic phonons lies in the range of 50 to 75 meV, giving a range of λ =0.05 to 0.12, so the contribution of phonons to the bandgap is estimated as 3% to 7%. I would expect BN to have stronger interaction with optical phonons, since the pattern of distortions around an electronic defect is consistent with optical modes (see Figure 1).
The BN gap is too wide for digital applications. Recently, it has become possible to manufacture silicene, an atomically thick layer of silicon with similar properties to graphene , so it may be possible to make GaAs or AlP analogues to BN. Smaller gaps could be available from those materials, which might be used to create tunable bandgaps for atomically thick transistors.
JPH is a lecturer from the Faculty of Science, Department of Physical Sciences, The Open University, Walton Hall, Milton Keynes, UK.
The author acknowledges EPSRC grant EP/H015655/1 for funding and useful discussions with A Ilie and A Davenport.
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