## 2 Calculation of the Density of Electronic States of Disordered tBLG in an External Field
To take into account the influence of an external field and disorder arising from the rotation of two layers of graphene relative to each other, using the Hamiltonian in the tight coupling model [31], the dispersion law is obtained as follows  
\[\epsilon_{k}=\pm\sqrt{\frac{e^{2}V^{2}}{4}+t_{0}^{2}+2((p^{2}-\omega^{2}))} \tag{2}\]  
where \(e\) is the electron charge, \(V\) is the external electrical field value, \(t_{0}\) is the integral of electron transition between sites, \(\omega=\left(\frac{4\pi\hbar c_{W}}{3\sqrt{3}g\omega_{0}}\right)^{2}\sin^{2}( \theta/2)\), \(v_{F}\) is the electron Fermi velocity, \(a_{0}=0.142\) nm is the distance between the nearest carbon atoms, \(p\) is the electron momentum, and \(\theta\) is the interlayer twist angle in tBLG. The expression for the density of electronic states of tBLG, taking into account the multiple elastic scattering of electrons by impurities and the structural inhomogeneities of the short-range order type, was calculated similarly [30]. In the first stage, the one-particle [PERSON]'s function, averaged over the disorder, is represented as follows  
\[\langle G\rangle=G_{0}+G_{0}^{2}\Sigma, \tag{3}\]  
where \(G_{0}=\frac{1}{\epsilon-\epsilon_{p}+t_{0}}\) is [PERSON]'s function. \[\sum=cU_{0}+c^{2}U_{0}^{2}G_{0}(p)+cU_{0}^{2}\int G_{0}(p_{1})dp_{1}+U_{0}^{2 }\int\langle|c(p-p_{1})|^{2}\rangle G_{0}(p_{1})dp_{1} \tag{4}\]  
is the self-energy part in momentum representation and \(U_{0}\) is the effective potential of multiple elastic scattering of the electrons on the defects. The first two terms in (3) determine only the contribution to the change in the chemical potential. The last two terms determine the electron relaxation time \(1/2\tau=-\text{Im}\,\Sigma\). Calculated from (3) with the replacement of (1), the relaxation time of electrons in tBLG has the form  
\[\tau=\frac{\tau_{0}}{1+BT^{1/2}}, \tag{5}\]  
where \(\tau_{0}=\frac{\hbar}{4\pi^{2}\iota_{0}^{2}\varsigma\upsilon_{0}}\) is the electron relaxation time by impurities, \(c\) is the impurity concentration, and \(\upsilon_{0}\) is the DOS at the Fermi level calculated for perfect tBLG, \(B=\frac{2\sqrt{2}\tau(1-c)\eta^{3/2}}{\upsilon_{0}N\hbar^{3}}k_{B}^{1/2}\alpha\). The averaged Green's function has the traditional form, but with relaxation time (4)  
\[\langle G\rangle=\frac{1}{\varepsilon-E_{p}+\frac{j}{2\tau}signe}, \tag{6}\]  
where \(E_{p}\) is the electron spectrum of disordered tBLG. Precisely, \(\tau\) is responsible for the peculiarities of the change in the one-electron properties of tBLG. As the change in the electronic spectrum is not decisive, it is assumed that \(\varepsilon_{p}\approx E_{p}\). This approach makes it possible to obtain an analytical expression for DOS, but without quantitatively taking into account the Fermi level shift. The contribution to the DOS of tBLG from electron scattering on structural inhomogeneities and areas with local short-range order can be calculated as \(\Delta\
u=-\frac{1}{\pi}\text{Im}Sp(\langle G\rangle-G_{0})\). The final calculated expression for the contribution to the DOS in scattering by impurities and short-range ordered regions in tBLG in an external field has the following form  
\[\Delta\upsilon=\frac{1}{2\tau\hbar}\frac{\sqrt{A}}{2\upsilon_{F}^{2}\left( \sqrt{A}-\varepsilon\right)}\left(p_{F}-\sqrt{\frac{\sqrt{A}\left(\left(\sqrt{ A}-\varepsilon\right)^{2}+\left(\frac{\hbar}{2\tau}\right)^{2}\right)}{2 \upsilon_{F}^{2}\left(\sqrt{A}-\varepsilon\right)}}arctg\left(\sqrt{\frac{2 \left(\sqrt{A}-\varepsilon\right)}{\sqrt{A}\left(\left(\sqrt{A}-\varepsilon \right)^{2}+\left(\frac{\hbar}{2\tau}\right)^{2}\right)}}pv_{F}\right)\right), \tag{7}\]  
where \(A=\frac{c^{2}\
u^{2}}{4}+t_{0}^{2}-2\omega^{2}\). The calculated contribution to the DOS (6) depends on the temperature, impurity concentration, short-range order parameter, interlayer twist angle, and external field via a complex relationship with the relaxation time and variable \(A\). Below, we analyze how each parameter affects the behavior of the DOS near the Fermi level.