Band structures and wavefunctions

Figure 4: PPP band structure with selected wavefunctions close to the band gap. Red and blue indicate positive and negative sign respectively

Figure 5: PBZ band structure with selected wavefunctions close to the band gap.

The electronic properties of these polymers are important for potential technological applications. In general, BN compounds tend to have larger energy gaps than their carbon analogues due to the polarity of the chemical bonds. This effect, which is observed when the band gaps of group-IV semiconductors are compared to those of III-V and II-VI semiconductors, also occurs for first row elements: e.g. cubic BN has a larger band gap than diamond, and graphitic BN is a wide gap insulator whereas graphite is a semi-metal. We therefore expect similar behaviour in the polymers. In Figs. 4 and 5 the band structures for PPP and PBZ from the Kohn-Sham eigenvalues obtained in our DFT calculations are shown, along with some of the wavefunctions for states close to the band gap. In particular, the lowest unoccupied (LUMO) state represents the wavefunction for electrons in the first excited state, while the highest occupied (HOMO) state represents the holes. It is known that DFT band calculations underestimate the band gap; for example for PPV we obtain a gap of 1.2 eV [27], whereas the known optical gap is 2.5 eV [30]. However, the error does not scale with the magnitude of the gap and it is usually less severe for large band gap compounds. Moreover, to quantitatively describe the absorption spectra of polymers, the electron-hole interaction, accounting for exciton formation, needs to be included [42,43,44,45,46,47,48,49,50]; this would lead to much more complicated and computationally demanding calculations. However, the comparison of these band structures should still capture the crucial qualitative features of the BN systems, as previously confirmed in the case of BN nanotubes [51]. Indeed, the band gaps of the BN polymers are much larger than those of the equivalent carbon systems; accounting for the correction in the gap underestimation and for the electron-hole interaction, this will result in band gaps in the UV. Wide gap materials are interesting per se, as testified for example by the rapidly developing research on GaN. Pure BN polymers, which are likely to have band gaps even larger than GaN, could be suitable candidate materials for ultraviolet detectors and emitters. Shorter wavelengths could be important for nanotechnology applications, where the effort is to build devices on increasingly smaller scales. While PPP clearly has a direct gap, the similar energies of the conduction states at $\Gamma$ and X of PBZ cannot rule out an indirect gap. For light polarized along the polymer chain, the direct HOMO-LUMO transitions at $\Gamma$ and at X are optically allowed in both PPP and PBZ, as demonstrated by the symmetry of the valence and conduction states at $\Gamma$ and X (see Figs. 4 and 5). Both PBZ and PVB show less dispersion in the states close the Fermi level as compared to PPP and PPV respectively. These features, together with the large gaps, will affect the electron-hole interaction, leading to differences in the absorption spectra of the BN and carbon polymers.