Metal-insulator transition in $8-Pmmn$ borophene under normal incidence of electromagnetic radiation

The energy spectrum for the problem of $8-Pmmn$ borophene's electronic carriers under normal incidence of electromagnetic waves is studied without the use of any perturbative technique. This allows us to study the effects of very strong fields. To obtain the spectrum and wave functions, the time-dependent Dirac equation is solved by using a frame moving with the space-time cone of the wave, i.e., by transforming the equation into an ordinary differential equation in terms of the wave phase, leading to an electron-wave quasiparticle. The limiting case of strong fields is thus analyzed. The resulting eigenfunctions obey a generalized Mathieu equation, i.e., of a classical parametric pendulum. The energy spectrum presents bands and a gap at the Fermi energy. The gaps are due to the space-time diffraction of electrons in phase with the electromagnetic field, i.e., electrons in borophene acquire an effective mass under strong electromagnetic radiation.

Figure 1

Borophene lattice for the crystal phase with space group $8-Pmmn$.

Figure 2

Plot of the energy dispersion $E\left(\mathbf{k}\right)$ as a function of $\mathbf{k}$ [see Eq. (2)] in the region around a Dirac cone. Notice the tilted anisotropy of the Dirac cone.

Figure 3

Stability chart as a function of the nondimensional parameters $a$ and $q$ for Mathieu solutions. The regions of stability (gray domains) and instability (white domains) are divided by the characteristic curves $a_n\left(q\right)$ and $b_n\left(q\right)$ (dashed lines). This spectrum is for the case of intense electric fields or long wavelengths. In this case, there is a constraint due to Eq. (22). For fixed energies $\epsilon$, this condition is equivalent to draw a set of parallel lines as shown in the figure, where $\mathrm{{\rm Y}}=\epsilon /\hslash \mathrm{\Omega }$. The line $a=-2q$, i.e., $\mathrm{{\rm Y}}=0$, divides the spectrum into two zones, the zone on the left side of this line is strictly not allowed, while the values on the right side are allowed. A simple analysis of this figure reveals bands separated by energy gaps. The order of the gaps is indicated.

Figure 4

Amplification of the energy spectrum for the case of long wave length near of $\mathrm{{\rm Y}}=0$, i.e., $\epsilon =0$. For a fixed field ($E_0=2$ V/m), the figure shows two energy first-order gaps for $\theta =0$ and $\theta =\pi /2$, respectively, and two energy bands. We observe that the gap size decrease while $\theta$ increase for $0\le \theta \le \pi /2$.