Here I have gathered a list of miscellaneous notes about the theory em-ployed byHOTBITand practical issues related to the usage ofHOTBIT(and ASE):
1. The ASE version 3.6.0. is preferred to be used withHOTBIT, since at the time HOTBIT was coded, it was the ASE version available. The newer versions might have some compatibility issues.
2. The units defined by Eqs. (2.1–2.8) are used internally by HOTBIT. But since HOTBIT is an ASE calculator, the units viewed from out-side are electronvolts and Ångströms. This has to be taken into ac-count with values ofU andrcov—they have to be given toHOTBITin Hartrees and Bohrs, respectively.
3. HOTBIT uses internally the reciprocal lattice defined by Eq. (3.6) in the k-point calculations. Consequently also Eqs. (3.7–3.15) apply.
But in ASE the reciprocal vectors are defined in such a way that biASE
= 1 ∀i. This induces relations kASE = √
3a/(4π)kHB for 2D h-BN structures (ais the lattice parameter) andkASE =l/(2π)kHB for BNNTs (lis the length of the tube). This has to be taken into account for example when plotting DFTB band structures to the same figure with band structures computed with ASE with other calculators.
6 Construction of BN-parametrization
Table 1: The used values ofU andrcov. The value forU has been calculated from IE andEAwith Eq. (2.93). In the parametrization theU in Ha andrcov in Bohr should be used as discussed in the precious section.
IE(eV)[31] EA(eV)[32] U rcov [33]
(eV) (Ha) (Å) (Bohr)
B 8.294 0.277 8.017 0.295 0.84 1.59
N 14.527 0.072 14.455 0.531 0.71 1.34
Table 1 features the values of U and rcov used in my parametrization.
I chose not to alter U at all, and I just employed the default U-values shown in this table. Also the used values for rcov read there. However, it afterwards turned out that I accidentally had used the rcov in eV in-stead of Bohrs. Luckily this mistake did not have any effect on the fi-nal parametrization since due to this I only thought I was dealing with 1/0.529 = 1.890 times larger x-parameters than I really was. In the fol-lowing I will however keep my erroneous x-parameter notation so now the x-parameters are defined through Vconf(|r|) = [|r|/(2·0.529·xrcov)]2 instead ofVconf(|r|) = [|r|/(2xrcov)]2, which is the Eq. (4.2).
6.1 The x-parameters
As discussed in Sec. 4.3.1, the quality of differentx-parameters should first be compared based on their effects on the band structures. More explic-itly, the general matching between the DFTB and reference (LDA-DFT) energy bands implies that the minimal basis set {ϕµ} is of good quality.
In this sense all the bands are equally important in seeking the good x-parameters. But on the other hand the band structures are not only a tool for finding the optimal x-parameters. They also are an application of the parametrization and in that sense the valence and conduction bands and the band gapEgap are particularly important. Thus, all in all, when judg-ing the quality ofx-parameters from the point of view of band structures, all the bands but particularly the conduction and valence bands and Egap
are important.
In my work I used as the only reference band structure system the h-BN
layer witha= 2.50Å [14] as the reference value for the lattice constant. All BN nanostructures are based on this layer and therefore good correspon-dence between the DFT and DFTB band structures probably ensures that they are good also for other BN nanostructures—such as BN nanotubes.
Hence making thex-parameter comparisons only based on the h-BN band structures should be enough.
I studied a quite broad range ofx = (xB, xN), since the sensible range for x-parameters is from∼ 0.6to infinity2—the lower bound is explained by the fact that too tight confinement results in{ϕµ}too different from{ϕfreeµ } and hence unphysical and the infinity limit just leads to {ϕµ} = {ϕfreeµ }. Now, the results of my experimentings with different x-parameters are presented in Fig. 5. Only a selected array of (xB, xN) in a bit narrow range is shown, but the general behavior should be well visible. It is seen that modifying thex-parameters around the default values does not have particularly drastic effects on the band structures and clearly for any pair (xB, xN)perfect match between the DFTB and LDA-DFT band structures is not found. But somex-parameter values are certainly better than others, so analyzing the effects of changing xB andxN to the h-BN BS properties can be done. And it seems that luckily these effects are somewhat system-atic and linear and the contributions fromxBandxNcan be approximately separated. This simplifies the process of choosing the final x-parameters as their effects can be considered independently.
When it comes to variation of xB, first of all the four lowest bands seem to mostly retain their shape, but increasing xB shifts these bands lower in the energy as a whole. As the result the valence band ends up in too low energy when compared to DFT. This increases the DFTB-Egap, since the conduction band is hardly altered by changes in xB. The conduction band is also too low in energy (even strongly and for anyxB), so lowering the valence band when increasing xB improves the DFTB-Egap—even for (xB, xN) = (3.0,2.0)we only haveEgap= 4.35eV whereas the experimental Egapis∼5eV. Considering all this, largerxBshould be preferred but also lowxBmight be an option if it provides betterVrep-functions.
Next, xN has even less influence on the h-BN BS. Bands from 2nd to 4th seem not to be altered by it nearly at all, but increasing it improves the first band. On the other hand, lowering it improves the conduction band around theΓ-point as well as the higher bands in general. Hence based on this analysis it is difficult to deduce the bestxNand it really does not rule
2Now with my mistake from∼1to infinity.
K Γ M K/K0
Figure 5:Band structure of the h-BN layer for several values for(xB, xN)—DFTB bands in solid coloured and LDA-DFT in grey dashed. There are 4 valence elec-tron positions in the h-BN unit cells (three from B and one from N) and thus the four lowest bands are the normally occupied valence bands. In singular, valence band refers to the highest of them (the 4th band) and conduction band refers to the band above it (the 5th band).
out any value for it from a sensible range. Thus the final value for xN is left to be decided by the quality ofVrep-parametrizations.
In the numerical calculation of the band structures I experimented with different settings with both DFTB and DFT (e.g. the Fermi width), but they had little influence on the results. With DFT I even tried more advanced xc-functionals than theVxcLDA, but the results were again similar.