Chirality-2 fermion induced anti-Klein tunneling in a two-dimensional checkerboard lattice

The quantum tunneling effect in a two-dimensional (2D) checkerboard lattice is investigated. By analyzing the pseudospin texture of the states in a 2D checkerboard lattice based on the low-energy effective Hamiltonian, we find that this system has a chiral symmetry with chirality equal to 2, although compared with regular chiral fermions, its pseudospin orientation does not vary uniformly. This suggests that the perfect reflection chiral tunneling, also known as the anti-Klein tunneling, is expected to appear in a checkerboard lattice as well. To verify the conjecture, we calculate the transmission probability and find that normally incident electron states can be perfectly reflected by the barrier with hole states inside and vice versa. Furthermore, we numerically calculate the tunneling conductance of the checkerboard nanotube using the recursive Green's function method. The results show that a perfect on-off ratio can be achieved by confining the energy of the incident states within a certain range. It also suggests that, by tuning the barrier, the checkerboard nanotube can work as a perfect band filter or tunneling field effect transistor, which transmits electrons selectively with respect to the pseudospin of the incident electrons.

Figure 1

(a) Checkerboard lattice with nearest- and next-nearest-neighbor hoppings. Two sublattices are labeled by solid and open circles. (b) Schematic diagrams of the band structure and variation of the electrostatic energy caused by the barrier. (c) The angle of pseudospin as a function of the angle of the wave vector: red for subband $s=1$ and blue for $s=-1$. (d) The textures of pseudospin at the Fermi surface with subband indexes $s=-1$ (blue) and $s=1$ (red) are denoted by the arrows of unit vectors.

Figure 2

The angular dependence of the transmission probability in the checkerboard lattice. We set the barrier potential (a) $V_s=-0.50,-0.28,-0.10$ and (b) $V_s=0.50,0.28,0.10$. The incident energy is set to be $E=-0.1$.

Figure 3

The tunneling conductance of the checkerboard lattice varies with the height of the barrier $V_s$. (a) The band structures of the tunneling barrier. The gray regions correspond to the situation that tunneling currents are almost entirely blocked. Incident wave of different energies are calculated: (b) $E_1=-0.1\in [-\mathrm{\Delta },0]$ and (c) $E_2=-0.375<-\mathrm{\Delta }$. Other parameters $M=10, \mathrm{\Delta }=\sqrt{3}(1-cos\frac{2\pi }{M})\approx 0.33$, and $D=10$. In (b) and (c), the blue solid lines represent the results from the transmission probability calculation (details shown in Appendix pp3) with the orthogonality assumption of wave functions with different $k_y$, and the red dots are results from the recursive Green's function.

Figure 4

(a) Band structure of the Hamiltonian with parameters obtained by density functional theory (DFT) fitting of $\tau$-type organic conductor. We set ${\tilde{k}}_y=\pi$. (b) and (c) Textures of pseudospin at the Fermi surfaces $E_f=0.03\mathrm{eV}$ (orange) and $E_f=-0.01\mathrm{eV}$ (blue), respectively.

Figure 5

The black curve is a typical Fermi surface of the checkerboard lattice for $t=t^{\prime }=-t^{\prime \prime }=1$. Specifically, here, $E^{\prime }=2.5$. The orange and blue lines correspond to $k_y=\frac{3\pi }{5}$ and $\frac{\pi }{5}$, respectively. The black curve and the orange line intersect at points $1^{\prime }$ and $2^{\prime }$, which represent propagating modes of the same group. The black curve and the blue line intersect at points 1, 2, 3, and 4, which represent propagating modes belonging to two groups.

Figure 6

The zero temperature linear tunneling conductance of the checkerboard lattice varies with the height $V_s$ and width $D$ of the barrier. Incident wave energy $E=-0.1\in [-\mathrm{\Delta },0]$. Other parameters $M=10, \mathrm{\Delta }=\sqrt{3}(1-cos\frac{2\pi }{M})\approx 0.33, D=8$ (top panel), $D=16$ (middle panel), and $D=24$ (bottom panel).