## 2 Material Parameters of the Structure
The photonic band gap (PBG) exists in the transmission spectrum \(T(f)\) in the region of the operating frequency \(f_{0}\) in the center of which a defect mode is observed--a narrow peak of almost complete transmission, associated with violation of the structure periodicity. In BMs, the transmission maximum corresponds to the center frequency of the non-transmission zone of a defect-free structure when the optical thicknesses of the layers are equal. The symmetry of the structure with respect to the location of the defect (in this case, the role of the defect is played by an unfilled cavity) leads to the maximum possible value of the transmittance in the defect mode. The spectral width of the defect mode essentially depends on the value of the layers' permittivity in the region of the defect. The width of the transmission miniband for a structure with an increased permittivity in the region of a defect is much wider than for a structure with a reduced value. An increase in the number of periods \(a\) in BMs leads to sharper band gaps and the narrowing of the spectral line of the defect mode, but does not affect its position in the transmission spectrum and the magnitude of the maximum \(T(f_{0})\). In this case, the degree of field localization in the region of the defect significantly increases. Thus, with an increase in the number of periods in BMs by two (one each to the left and to the right of the cavity), the square of the field amplitude at the maximum doubles, i.e., \(|E_{max}(a+1)|^{2}\simeq 2|E_{max}(a)|^{2}\). Then, we will consider an MCR with a cavity filled with a periodic structure of alternating layers of dielectric (or semiconductor) and graphene monolayer with thicknesses \(\ell_{d}\) (or \(\ell_{s}\)) and \(\ell_{g}\). The material parameters of each layer in the studied frequency range are scalar-tensor quantities: for the dielectric and graphene, the permittivities \(\varepsilon_{d}\) and \(\varepsilon_{g}\); for semiconductor \(\varepsilon_{s}\), the magnetic permeability of all layers in the studied range can be considered equal to unity. Such a one-dimensional structure has the properties of a uniaxial crystal with a symmetry axis perpendicular to the layer interfaces (\(OZ\) axis). The transfer matrix of the insert has the form \(N_{3}=(N_{g}N_{u})^{b}N_{g}\), where \(N_{u}=N_{d,s}\) for dielectric and semiconductor layers, parameter b specifies the number of periods. In this case, the transfer matrices \(N_{d}\) and \(N_{s}\) have the form of a matrix (2) with the corresponding permittivities. For semiconductor layers in the case of an external magnetic field oriented along the \(OX\) axis, the nonzero components of the semiconductor permittivity tensor \(\varepsilon_{yy}=\varepsilon_{zz}=\varepsilon\), \(\varepsilon_{xx}=\varepsilon_{\parallel}\), \(\varepsilon_{yz}=-\varepsilon_{zy}=i\varepsilon_{a}\) depend on frequency and field as follows [34]:  
\[\varepsilon=\varepsilon_{l}\bigg{(}1-\frac{\omega_{p}^{2}\omega_{v}}{\omega( \omega_{v}^{2}-\omega_{c}^{2})}\bigg{)},\quad\varepsilon_{\parallel}= \varepsilon_{l}\bigg{(}1-\frac{\omega_{p}^{2}}{\omega\omega_{v}}\bigg{)}, \quad\varepsilon_{a}=\frac{\varepsilon_{l}\omega_{p}^{2}\omega_{c}}{\omega( \omega_{v}^{2}-\omega_{c}^{2})}, \tag{4}\]  
where \(\omega_{p}=\sqrt{4\pi e^{2}n_{0}/m^{*}\varepsilon_{l}}\) and \(\omega_{c}=eH_{0}/m^{*}c\) are the plasma and cyclotron frequencies of the semiconductor, \(\varepsilon_{l}\) is the lattice part of the permittivity, \(e\) is the charge of an electron, \(n_{0}\) and \(m^{*}\) are the concentration and the effective mass of the carriers, respectively, \(\omega_{v}=\omega+iv\) and \(\
u\) is a relaxation parameter. The eigenwaves of the semiconductor medium are \(TE\) and \(TM\) waves at the specified orientation of the magnetic field and the transverse propagation of the electromagnetic wave (\(\mathbf{k}\perp\mathbf{H}_{0}\)). Further analysis will be carried out for a \(TM\)-type wave, since it is a wave of this type that is controlled by an external field.