## 3 Regularities of Variation in the DOS of Perfect and Disordered tBLG  
The contribution to the DOS (6) is calculated at the following parameters: the interlayer twist angle 5-20\({}^{\circ}\), temperature 10-100 K, defect concentration of 0-0.05 for the impurity contribution and 0.05-0.15 for the short-range order contribution, and \(U_{0}=0.05\) eV. Short-range order parameters are calculated using the developed software and vary from \(-0.23\) to \(0.4\), which corresponds to different mutual positions and concentrations of foreign atoms in the tBLG structure. The sign of the short-range order parameter is negative if foreign atoms predominate in the first coordination sphere and positive if they are located in the second coordination sphere. Foreign atoms have different positions on account of the interaction with various substrates, functionalization, and other types of postprocessing of bigraphene. Regularities of the DOS behavior of tBLG near the Fermi level are studied at various angles, impurity concentrations and configurations, temperature, and external field values. The research results are given below. In the first stage, we study how point defects (\(\alpha=0\), \(c\
eq 0\)) affect the behavior of the DOS near the Fermi level in tBLG (see Figure 1). It is shown that point defects increase the DOS near the Fermi level (Figure 1a), which agrees with the results of the work dealing with the effect of vacancy defects on the electronic structure [32; 33]. The DOS near \(\varepsilon_{F}\) increases with the growing concentration of point defects and twist angle, and decreases in a stronger external field (Figure 1b). [PERSON] et al. [34] studied zero-field dI/dV spectra of bilayer graphene with \(\theta=3.6^{\circ}\) on the substrates made of silicon carbide (SiC) and highly oriented pyrolitic graphite (HOPG). According to our calculations, the resulting corrugation of the surface at this twist angle leads to the interaction of AA superstructures (amounting to 10% of the surface area) with the substrates. In this case, the distance from the tBLG to the HOPG substrate is about two coordination spheres, and to the SiC substrate it is one coordination sphere. Therefore, contribution (6) is calculated at the following parameters: \(c=0.1\) and \(\alpha=-0.23\) for the SiC substrate, and \(\alpha=0.4\) for the HOPG substrate. The calculated dependences \(\Delta\
u(\varepsilon)\) and the corresponding experimental data are shown in Figure 2a. The obtained theoretical curves are in qualitative agreement with the experimental data (Figure 2b). Figure 2: Contribution to the DOS (6) calculated at \(\theta=3.6^{\circ}\), \(c=0.1\), (**a**); zero-field dI/dV spectra of twisted graphene on the HOPG (red) and SiC (black) substrates at the same twist angle \(\sim 3.6^{\circ}\) (modified from [34]) (**b**). The temperature dependence of the DOS near the Fermi level is radically different for the cases \(\alpha>0\) and \(\alpha<0\) at the fixed \(\theta\). Figure 3 exhibits the calculated contribution to the DOS (3) at the three temperatures: 10 K (blue), 100 K (red line), and 300 K (blue line). As can be seen, at a higher temperature, the DOS near the Fermi level increases at \(\alpha>0\) (Figure 3a) and decreases at \(\alpha<0\) (Figure 3b). Due to the negative contribution of electron scattering by short-range order regions at a high temperature, a gap opening is possible in the DOS. At low temperatures (up to 100 K), no gap of the DOS opens in tBLG at any negative short-range order parameters and concentrations in the zero external field. The possibility of variation in the DOS near \(\varepsilon_{F}\) with temperature is confirmed experimentally [35]. To determine the influence of the external field on the behavior of the DOS near \(\varepsilon_{F}\), we consider contributions (6) at \(\alpha<0\) and \(\alpha>0\) (Figure 4). With foreign atoms in the first coordination sphere (\(\alpha<0\)), the negative contribution (6) can cause a gap opening in the DOS in an external field. The higher V, the wider the gap \(\Delta\).