# Floquet Modification of the Bandgaps and Energy Spectrum in Flat-Band Pseudospin-1 Dirac Materials
## 4 Results and Discussion  
Our main goal is to investigate the band structure of the electron dressed states for the \(\alpha-\mathcal{T}_{3}\) materials with a finite gap, i.e., to understand how the existing gaps are modified or renormalized due to the elliptically polarized dressing field. First, we find that according to Equation (34), which demonstrates how in principle the bandgap is created for \(\lambda_{0}\ll 1\), the structure of the gap induced by the dressing field is, like the existing gap, given by Equation (13). However, an important difference between the two types of gap is that the irradiation-induced gap directly depends on the valley index \(\tau\), and it is opposite for the \(K\) and \(K^{\prime}\) valleys. Therefore, the results of the two valleys are exactly opposite, and the same initial gap could be either decreased or increased depending on the value of \(\tau=\pm 1\). This is exactly what we see when comparing the upper and the lower panels of Figures 3-5. In our calculations, the following system of units was employed. All the energy values, including bandgap \(\Delta_{0}\), are measured in terms of a typical graphene Fermi energy \(E_{F}^{(0)}\), which is obtained as \(E_{F}^{(0)}=\hbar v_{F}k_{F}^{(0)}\). Here, \(k_{F}^{(0)}\) is the corresponding Fermi momentum, which is also a reciprocal to our unit of length \(1/[L_{0}]\); \(n_{0}\) and \(v_{F}\) denote the areal electron doping density and the Fermi velocity, respectively. Thus, for a typical density \(n_{0}\sim 10^{11}\) cm\({}^{-2}\), we obtain \(k_{F}^{(0)}\simeq 8.0\times 10^{7}\) m\({}^{-1}\), \(L_{0}\simeq 10^{-8}\) m and \(E_{F}^{(0)}\sim 7.5\times 10^{-21}\,J\simeq 50\) meV. Our numerical results, which are presented in Figure 3, demonstrate how a finite bandgap is opened for an initially gapless \(\alpha-\mathcal{T}_{3}\) material, as it is always expected due to the circularly polarized dressing field. In contrast, the change in the existing bandgap presented in Figure 4 substantially depends on the electron-light coupling parameter \(\lambda_{0}\), the initial bandgap \(\Delta_{0}\), phase \(\phi\), and the valley index \(\tau\). We note that the bandgaps and the dispersions of the dressed state subbands could be modified in completely different ways, depending on the material parameters of a specific \(\alpha-\mathcal{T}_{3}\) lattice. Figure 4: (Color online) The \(k_{x}\)-dependence of the energy dispersions \(\varepsilon_{d}(\lambda_{0}|k_{x},k_{y}=0)\) for \(\alpha-\mathcal{T}_{3}\) materials. Each panel corresponds to a finite bandgap \(\Delta_{0}=1.0\) and a fixed phase \(\phi=\pi/8\) and different values of the coupling constant \(\lambda_{0}\), as labeled. All the upper plots **(a–c)** are related to the \(K\) valley with \(\tau=1\), and the lower ones **(d–f)**—to the \(K^{\prime}\) valley—with \(\tau=-1\). Figure 3: (Color online) The \(k_{x}\)-dependence of the energy dispersions \(\varepsilon_{d}(\lambda_{0}|k_{x},k_{y}=0)\) for \(\alpha-\mathcal{T}_{3}\) materials. Each panel corresponds to a zero bandgap \(\Delta_{0}=0\), a fixed phase \(\phi=\pi/8\), and a different value of the coupling constant \(\lambda_{0}\), as labeled. All the upper plots **(a–c)** are related to the \(K\) valley with \(\tau=1\), and the lower ones **(d–f)**—to the \(K^{\prime}\) valley—with \(\tau=-1\). Importantly, we also reveal a substantial dependence of the irradiation-induced bandgap on the phase \(\phi\). For the specific situation demonstrated in Figure 5 with the initial bandgap \(\Delta_{0}=1.0\), we see that both gaps are significantly increased for the larger values of \(\phi\), just like the equivalence (the differences) between the two gaps. The dressing field of elliptical polarization with \(\beta<1\) (in contrast to circularly polarized light with \(\beta=1\)) is expected to induce finite anisotropy and an angular dependence of the initially isotropic dispersions with a circular constant-energy cut.