Frequency-dependent magneto-optical conductivity in the generalized $\alpha -T_3$ model

We have studied a generalized three-band crossing model in 2D, the generalized $\alpha -T_3$ lattice, ranging from the pseudospin-1 Dirac equation through a quadratic+flat band touching to the pseudospin-1/2 Dirac equation. A general method is presented to determine the operator form of the Green's function, being gauge and representation independent. This yields the Landau level structure in a quantizing magnetic field and the longitudinal and transversal magneto-optical conductivities of the underlying system. Although the magneto-optical selection rules allow for many transitions between Landau levels, the dominant one stems from exciting a particle from/to the flat band to/from a propagating band. The Hall conductivity from each valley is rational (not quantized at all), in agreement with Berry phase considerations, though their sum is always integer quantized.

Figure 1

The dice lattice with $t_1$ and $t_2$ hopping amplitude along the red dashed and blue solid lines. The on-site energy of the sixfold connected site $B$ is ${\epsilon }_0$. There are three atoms $A,B$, and $C$ in each unit cell.

Figure 2

The energy dispersion is sketched in the low-energy limit of the generalized $\alpha -T_3$ model, Eq. (5) for ${\epsilon }_0=0$ (left) and ${\epsilon }_0>0$.

Figure 3

Allowed transitions for different values of the Fermi energy.

Figure 4

The real part of the longitudinal conductivity (in units of $e^2/h$) as a function of the frequency $\omega$ (in units of K) for Fermi energy $\mu =500$ K [blue line, (a)] which is in the gap, $\mu =730$ K [black line, (b)] which lies between the Landau levels $n=0$ and $n=1$ in the $K$ valley and in the gap in the $K^{\prime }$ valley, and $\mu =850$ K [red line, (c)], which is between the Landau levels $n=0$ and $n=1$ for both valleys. The parameters are $T=10$ K, ${\epsilon }_0=500, {\beta }_1=350$ K, ${\beta }_2=450$ K, and $\eta =5$ K. For the first few peaks, the corresponding transitions are assigned in the form of $|n,\zeta \rangle \to |n^{\prime },{\zeta }^{\prime }\rangle$.

Figure 5

The real part of the longitudinal conductivity (in units of $e^2/h$) as a function of the inverse square root of magnetic field $B$ (here $B$ is in units of $\mathrm{T}$) for (a) ${\beta }_1={\beta }_2=400\mathrm{K}\sqrt{B}$, (b) ${\beta }_1=300\mathrm{K}\sqrt{B}$ and ${\beta }_2=400\mathrm{K}\sqrt{B}$ for the blue solid line, ${\epsilon }_0=100$ K for the red dotted line. The parameters are $T=10$ K, $\mu =50$ K, $\omega =200$ K, $\eta =5$ K, and ${\epsilon }_0=0$ in both cases.

Figure 6

The imaginary part of the transversal conductivity (in units of $e^2/h$) as a function of the frequency $\mathrm{\Omega }=\hslash \omega /k_{\mathrm{B}}$ (in units of K) (a) for Fermi energy $\mu =50$ K lying between the flat band and the first LL (blue solid line) and $\mu =500$ K which is between the first and second LL (red dashed line), (b) for $\mu =50$ K and for two sets of hopping parameters: ${\epsilon }_0={\beta }_1=0, {\beta }_2=423$ K corresponding to graphene with $B=1$ T magnetic field (blue solid line), and ${\epsilon }_0=0, {\beta }_1={\beta }_2=423$ K related to the Dirac-Weyl model for $s=1$ (red dashed line). The parameters are $T=10$ K and $\eta =5$ K in both cases.

Figure 7

The imaginary part of the transversal conductivity (in units of $e^2/h$) as a function of the frequency $\mathrm{\Omega }=\hslash \omega /k_{\mathrm{B}}$ (in units of K), for hopping amplitudes ${\epsilon }_0=0, {\beta }_1=380$ K, ${\beta }_2=420$ K (blue solid line), and for ${\epsilon }_0=0, {\beta }_1=300$ K, ${\beta }_2=500$ K (red dashed line). The parameters are $T=10$ K and $\eta =5$ K.

Figure 8

The transversal conductivity (in units of $e^2/h$) in dc limit $(\omega =0)$ as a function of the Fermi energy. Insets (upper right and lower left) show the contributions from the $K$ and $K^{\prime }$ valleys, respectively. The parameters are $T=0.01$ K, $\eta =5$ K, ${\epsilon }_0=0, {\beta }_1=300$ K, and ${\beta }_2=500$ K.

Figure 9

The transversal conductivity (in units of $e^2/h$) in dc limit $(\omega =0)$ as a function of the Fermi energy (spin degeneracy factor included!). Insets (upper right and lower left) show the contributions from the $K$ and $K^{\prime }$ valleys, respectively. The parameters are $T=0.01$ K, $\eta =5$ K, ${\epsilon }_0=600$ K, ${\beta }_1=300$ K, ${\beta }_2=500$ K for the red dotted line and ${\epsilon }_0=600$ K, ${\beta }_1={\beta }_2=412$ K for the blue solid line, so that ${\beta }_1^2+{\beta }_2^2$ has the same value for the two cases.