Nonequilibrium transport in the pseudospin-1 Dirac-Weyl system

Recently, solid state materials hosting pseudospin-1 quasiparticles have attracted a great deal of attention. In these materials, the energy band contains a pair of Dirac cones and a flatband through the connecting point of the cones. As the “caging” of carriers with a zero group velocity, the flatband itself has zero conductivity. However, in a nonequilibrium situation where a constant electric field is suddenly switched on, the flatband can enhance the resulting current in both the linear and nonlinear response regimes through distinct physical mechanisms. Using the ($2+1$)-dimensional pseudospin-1 Dirac-Weyl system as a concrete setting, we demonstrate that, in the weak field regime, the interband current is about twice larger than that for pseudospin-$\frac{1}{2}$ system due to the interplay between the flatband and the negative band, with the scaling behavior determined by the Kubo formula. In the strong field regime, the intraband current is $\sqrt{2}$ times larger than that in the pseudospin-$\frac{1}{2}$ system, due to the additional contribution from particles residing in the flatband. In this case, the current and field follow the scaling law associated with Landau-Zener tunneling. These results provide a better understanding of the role of the flatband in nonequilibrium transport and are experimentally testable using electronic or photonic systems.

Figure 1

Interband current in pseudospin-1 and pseudospin$\frac{1}{2}$ systems. (a) Evolution of the total current to electric field ratio $\tilde{J}/\tilde{E}$ with time $\tilde{t}$ for pseudospin-1 and $\frac{1}{2}$ systems for a fixed electric field $\tilde{E}=0.0004$, where the dashed lines denote the theoretical values ${\pi }^2/2$ and ${\pi }^2/4$ for the pseudospin-1 and pseudospin-$\frac{1}{2}$ systems, respectively. The yellow and green lines represent the respective numerical results. (b) The total current $\tilde{J}$ vs the electric field $\tilde{E}$ at time $\tilde{t}=2$ for the two systems. Comparing with the pseudospin-$\frac{1}{2}$ system, the interband current in the pseudospin-1 system is greatly enhanced.

Figure 2

Origin of interband current in the pseudospin-1 system. (a) Ratio between interband currents from the pseudospin-1 and pseudospin-$\frac{1}{2}$ systems as a function of time for electric field strength $\tilde{E}=0.0004$, and (b) current ratio vs $\tilde{E}$ for fixed time $\tilde{t}=2$. The black dashed lines are theoretical results, and the red and blue lines are for flatband and positive band, respectively. These results indicate that, for the pseudospin-1 system, the flatband is the sole contributor to the interband current.

Figure 3

Interband current distribution in momentum space: (a) pseudospin-1 and (b) pseudospin-$\frac{1}{2}$ systems. The time and electric field strength are $\tilde{t}=2$ and $\tilde{E}=0.0128$, respectively.

Figure 4

Enhancement of intraband current in the strong electric field regime. Intraband current and contributions from distinct bands (a) vs time for $\tilde{E}=0.8192$, where the black dashed lines represent the analytical values $2\left(\sqrt{2}-1\right)$, 2, $2\sqrt{2}$ (from bottom) and (b) vs electric field at time $\tilde{t}=10$ (for six values of the electric field: $\tilde{E}=0.2048$, 0.4096, 0.8192, 1.6384, 3.2768).

Figure 5

Further evidence of enhancement of intraband current in the pseudospin-1 system. (a) The ratio of the intraband currents in the pseudospin-1 and pseudospin-$\frac{1}{2}$ systems vs time $\tilde{t}$ for $\tilde{E}=0.8192$. (b) The current ratio vs $\tilde{E}$ for $\tilde{t}=10$.

Figure 6

Numerical evidence of pair creation mechanism for the intraband current. The ratio of particle number distribution for pseudospin-1 and pseudospin-$\frac{1}{2}$ systems (a) vs time $\tilde{t}$ for $\tilde{E}=0.8192$ and (b) vs $\tilde{E}$ for $\tilde{t}=10$.

Figure 7

Current density distribution in momentum space. (a), (b) For pseudospin-1 and pseudospin-$\frac{1}{2}$ systems, respectively, the distributions of the current density in momentum space for $\tilde{t}=20$ and $\tilde{E}=0.0512$. When the momentum gap value ${\tilde{p}}_y$ is large, the flatband can enhance the current.

Figure 8

Current vs electric field of pseudospin-1 system for $\tilde{t}=5$. As the magnitude of the external electrical field is increased, the dominant contribution to the total current changes from interband to intraband, and the algebraic scaling exponent of the current-field relation changes from 1 to 1.5.