Abstract
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 electronphonon coupling λ =1, indicating that a proportion of the measured BN bandgap may have a phonon origin.
Keywords:
Boron nitride; Electronphonon interactions; Semiconductors; Twodimensional materials; GrapheneBackground
The need for bandgaps in graphene on electronvolt scales has led to a number of proposals, such as the use of bilayer graphene [1], creation of nanoribbons [2], 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 [5]. This may lead to a complimentary method of manipulating bandgaps to make digital transistors.
In low dimensional materials, strong effective electronelectron interactions can be induced via an interaction between electrons confined to a plane and phonons in a polarizable neighboring layer [6]. The theory has shown that similar interactions account for the transport properties of graphene on polarizable substrates [7] and that sandwiching graphene between polarisable superstrates and gap opening substrates can cause gap enhancement [8]. This paper examines similar gap changes in atomically thin BN due to interactions mediated through substrates.
Methods
Atomically thick hexagonal BN (hBN) has similar chemistry to graphene: bonding occurs
through sp_{2} hybridization, and electrons with energies close to the chemical potential are in
unhybridized π orbitals [9]. A key difference is that the electronic charge is not completely screened by the
sp_{2} hybridization, shifting π orbitals by
Δ _{n }= + Δ on N sites, and −Δ on B sites. This shift is the dominant cause of a gap of order 2Δ . Tightbinding fits
to results from abinitio simulations of monolayer BN have established the hopping t =2.33 eV [10], with an estimate of Δ =1.96 eV=0.84 t . The experiments indicate larger gaps: bulk hBN has 5.971 eV [11], and monolayer hBN has a gap of 5.56 eV [12] corresponding to Δ =2.78 eV=1.20 t . There is significant variation in phonon energies,
The Hamiltonian terms are shown schematically in Figure 1 (a).
Figure 1 . Bn substrate system and interactions in a monolayer of BN.(a) BN substrate system annotated with interactions. Electronphonon interactions between the BN layer and substrate are poorly screened, and large interactions of strength f _{n }(m ) are possible. Ions in the substrate oscillate with frequency Ω. N sites have energy + Δ and B sites −Δ , opening a gap. The attractive phononmediated electronic interaction f binds electrons onto the same site, effectively enhancing the gap. (b) Interactions in a monolayer of BN. Red circles represent N atoms and black circles B atoms. Light blue arrows represent distortions expected from an excess of charge on the site labelled e .
For simplicity, the Holstein electronphonon interaction was used,
Results and discussion
The low order perturbation theory is applicable for low phonon frequency and weak coupling. A set of gap equations was derived by symmetrizing the self energy,
The local approximation used here is a good starting point because the modulated potential
Δ is large, and electrons are well localized. Offdiagonal terms do not feature in
the lowest order perturbation theory for the Holstein model since the interaction
is site diagonal. Z _{n } is the quasiparticle weight and
The full Green function can be established using Dyson’s equation
Substituting the expression for the Green function into the lowest order contribution to the self energy,
Here, the phonon propagator,
where the full gap is
Gap and quasiparticle weight functions only have a weak Matsubara frequency dependence
(<0.3% for
Figure 2 . Modification of the BN bandgap.(a) The gap enhancement depends mainly on λ , is weakly dependent on Δ and shows almost no change with Ω. Calculations are made
for Δ =t corresponding to a BN gap of 2Δ =4.66 eV, Δ =1.20t (2Δ =5.6 eV), and Δ =0.84t (2Δ =3.92 eV, the tight binding fit from reference [10]). t =2.33 eV,
Conclusions
A theory for the modification of BN bandgaps by interaction with phonons was presented here. It is of interest to make a comparison between the bandgaps of bulk hBN, nanotubes, monolayer hBN, and the theory presented here. Measured bandgaps of bulk hBN are of between 5.8 eV [15] and 5.971 eV [11], 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) [16]. On the other hand, Song et al. [12] 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
electronphonon 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 [17]. Extrapolating the interaction and taking a meanfield average (assuming mean momentum
magnitude of 4π /9a ), the electronphonon coupling can be estimated as
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 [18], 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.
Competing interest
The author declares that he has no competing interest.
Author’s information
JPH is a lecturer from the Faculty of Science, Department of Physical Sciences, The Open University, Walton Hall, Milton Keynes, UK.
Acknowledgements
The author acknowledges EPSRC grant EP/H015655/1 for funding and useful discussions with A Ilie and A Davenport.
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