Electron transitions for Dirac Hamiltonians with flat bands under electromagnetic radiation: Application to the $\alpha -{\mathcal{T}}_3$ graphene model

In a system with a Dirac-like linear dispersion there are always states that fulfill the resonance condition for electromagnetic radiation of arbitrary frequency $\mathrm{\Omega }$. When a flat band is present, two coexistent kinds of resonant transitions are found. Considering the $\alpha -{\mathcal{T}}_3$ graphene model as a minimal model with a flat band and Dirac cones and describing the dynamics using the interaction picture, we study the band transitions induced by an external electromagnetic field. We find that transitions depend upon the relative angle between the electron momentum and the electromagnetic field wave vector. For parllel incidence, the transitions are found using Floquet theory, while for other angles perturbation theory is used. In all cases, the transition probabilities and the frequencies are found. For the parallel momentum, no symmetry is broken by the field, and light does not change the spectrum, while for some limit special cases of the parameter $\alpha$ or by charge doping, the system behaves as a three-level or two-level Rabi system. All these previous results were compared with numerical simulations. Good agreement was found between both. The obtained results show a rich system in which different kinds of transitions coexist.

Figure 1

Sketch of the $\alpha -{\mathcal{T}}_3$ lattice. When $\alpha =0$, (${\mathcal{C}}_{\alpha }=1$, ${\mathcal{S}}_{\alpha }=0$) results in the honeycomb lattice resembling monolayer graphene. In contrast, for $\alpha =1$, (${\mathcal{C}}_{\alpha }={\mathcal{S}}_{\alpha }=1/\sqrt{2}$) leads to the well-studied dice lattice with pseudospin 1 [25].

Figure 2

Band structure of the $\alpha -{\mathcal{T}}_3$ lattice around the Dirac point K. The light-blue Dirac cone corresponds to the CB ($E_3$), light green shows the FB ($E_0$), and light orange shows the VB ($E_1$). When an electromagnetic field with a given constant frequency $\mathrm{\Omega }$ is introduced, direct interband transitions denoted by light-red arrows are allowed. (a) shows the transitions for the resonant frequency $\omega \approx \mathrm{\Omega }$ (corresponding to ${\kappa }_x^2+{\kappa }_y^2\approx 1$, as indicated by the gray circle), which is associated with two simultaneous transitions from the VB to the FB and from the FB to the CB. In (b), we show the transition for the resonant frequency $\omega \approx \mathrm{\Omega }/2$ (corresponding to ${\kappa }_x^2+{\kappa }_y^2\approx 1/4$, as indicated by the gray circle), which is associated with the transitions from the VB to the CB.

Figure 3

Occupation probabilities $|{\chi }_1{\left(t\right)|}^2$ (black line), $|{\chi }_2{\left(t\right)|}^2$ (red line), and $|{\chi }_3{\left(t\right)|}^2$ (blue line) as a function of time for the resonant frequency $\omega \approx \mathrm{\Omega }$. The dashed gray lines are the results obtained for the three-level Rabi problem considering only resonant terms. (a) shows the numerical results for $\alpha =1.0$ and ${\mathrm{\Omega }}_w=1.72$ GHz. (b) shows the behavior for $\alpha =0.2$ with ${\mathrm{\Omega }}_w=0.66$ GHz. The initial states in both panels are given by ${\chi }_1\left(0\right)=1$. Note that although ${\mathrm{\Omega }}_w$ is different in both panels, the periodicity is determined by ${\mathrm{\Omega }}_w$. Notice how the case $\alpha =1$ corresponds to the three-level Rabi problem.

Figure 4

Occupation probabilities $|{\chi }_1{\left(t\right)|}^2$ (black line), $|{\chi }_2{\left(t\right)|}^2$ (red line), and $|{\chi }_3{\left(t\right)|}^2$ (blue line) as a function of time for the resonant frequency $\omega \approx \mathrm{\Omega }$. The dashed gray lines are the results obtained for the three-level Rabi problem considering only resonant terms. (a) shows the numerical results for $\alpha =1.0$ and ${\mathrm{\Omega }}_w=3.45$ GHz. (b) shows the behavior for $\alpha =0.2$ with ${\mathrm{\Omega }}_w=1.32$ GHz. The initial states in both panels are given by ${\chi }_2\left(0\right)=1$. Note that although ${\mathrm{\Omega }}_w$ is different in the two panels, the periodicity is determined by ${\mathrm{\Omega }}_w$. Notice how the case $\alpha =1$ corresponds to the three-level Rabi problem.

Figure 5

Occupation probabilities $|{\chi }_1{\left(t\right)|}^2$ (black line), $|{\chi }_2{\left(t\right)|}^2$ (red line), and $|{\chi }_3{\left(t\right)|}^2$ (blue line) as a function of time for the resonant frequency $\omega \approx \mathrm{\Omega }/2$. The dashed gray lines are the results obtained for the two-level Rabi problem considering only resonant terms. (a) shows the numerical results for $\alpha =0.0$ and ${\mathrm{\Omega }}_s=3.45$ GHz. (b) shows the behavior for $\alpha =0.8$ with ${\mathrm{\Omega }}_s=0.75$ GHz. The initial states in both panels are given by ${\chi }_1\left(0\right)=1$. Note that although ${\mathrm{\Omega }}_s$ is different in the two panels, the periodicity is determined by ${\mathrm{\Omega }}_s$. Notice how the case $\alpha =0$ corresponds to the two-level Rabi problem.