Barrier transmission of Dirac-like pseudospin-one particles

We address the problem of barrier tunneling in the two-dimensional ${\mathcal{T}}_3$ lattice (dice lattice). In particular, we focus on the low-energy, long-wavelength approximation for the Hamiltonian of the system, where the lattice can be described by a Dirac-like Hamiltonian associated with a pseudospin one. The enlarged pseudospin $S=1$ (instead of $S=\frac{1}{2}$ as for graphene) leads to an enhanced “super” Klein tunneling through rectangular electrostatic barriers. Our results are confirmed via numerical investigation of the tight-binding model of the lattice. For a uniform magnetic field, we discuss the Landau levels and we investigate the transparency of a rectangular magnetic barrier. We show that the latter can mainly be described by semiclassical orbits bending the particle trajectories, qualitatively similar as it is the case for graphene. This makes it possible to confine particles with magnetic barriers of sufficient width.

Figure 1

(a) The ${\mathcal{T}}_3$ lattice in the $x$$y$ plane, characterized by translation vectors ${\mathbf{v}}_1=(3/2;-\sqrt{3}/2)a_0$ and ${\mathbf{v}}_2=(3/2;\sqrt{3}/2)a_0$, with lattice constant $a_0$. Rim sites A and B (open circles and squares, respectively) have a lower connectivity of 3 compared to 6 for the hub site (solid circles). The ${\mathcal{T}}_3$ lattice can be viewed as two nested HCLs (emphasized by the two shaded hexagons). (b) The energy spectrum of the bulk ${\mathcal{T}}_3$ lattice. The first Brillouin zone is indicated by the dashed line on the flat band.

Figure 2

(a) Sketch of the scattering region with an electrostatic barrier of width $d$. (b) Wave vectors corresponding to the different plane-wave components in ${\psi }_{\mathrm{I}}$, ${\psi }_{\mathrm{II}}$, and ${\psi }_{\mathrm{III}}$; see Eqs. (9, 10, 11) for $E>V_0$. (c) Scattering trajectory in the special case $E=V_0/2$. Note that the opposite orientation of $\mathbf{q}$ in the barrier region is due to propagation of holes [$(E-V_0)<0$] in II, compared to electrons ($E>0$) in I and III.

Figure 3

Polar plot of the transmission probability $T$ as a function of the injection angle $\varphi$ for different values of the energy $E$ for (a) the ${\mathcal{T}}_3$ lattice and for (b) the HCL lattice. The width of the barrier is $d{V}_0=60$ in both panels. The red dashed line is for $E=0.1V_0$, the blue dotted line is for $E=0.4V_0$, the green solid line is for $E=0.5V_0$, and the black dot-dashed line is for $E=0.95V_0$. For the ${\mathcal{T}}_3$ lattice at energy $E=0.5V_0$ we get perfect transmission, independent of the incident angle $\varphi$.

Figure 4

Comparison of numerical transport simulations and the analytical result for the continuum limit. Panels (a) and (b) show the geometries used in the numerical simulations, with the potential step along either (a) a zigzag direction or (b) an armchair direction. Panels (c)–(f) show a comparison of the transmission $T$ as a function of the incident angle $\varphi$ as computed numerically from the tight-binding model (black squares, armchair; blue crosses, zigzag) and from the continuum result (13) [red line, left out for clarity in (f), where there is only perfect forward scattering at $\varphi =0$]. The parameters used are $E=0.01t$, $d=1200\sqrt{3}{a}_0$, $d_{\mathrm{s}}=\sqrt{3}{a}_0$, and (c) $V_0=0.02t$, (d) $V_0=0.018t$, (e) $V_0=0.015t$, and (f) $V_0=0.01t$.

Figure 5

(a) Transmission as a function of incident angle $\varphi$ through a smoothed potential step along a zigzag (black line) and armchair direction (red line). Parameters are $E=0.1t$, $V_0=0.2t$, $d=200\sqrt{3}{a}_0$, and $d_{\mathrm{s}}=5.5\sqrt{3}{a}_0$. (b) Probability for a valley flip in the scattering process (given by the total probability of transmission or reflection into the opposite valley, $T_{\mathrm{inter}}+R_{\mathrm{inter}}$) for a smoothed potential step along the armchair direction at the incident angle $\varphi ={70}^{\circ }$ as a function of $E$. The remaining parameters are as in (a).

Figure 6

The transmission probability $T$ as a function of the injection angle for a magnetic barrier of width $d$ and at the energy $\mathcal{E}$ for the ${\mathcal{T}}_3$ lattice. In panel (a) the parameters are $\mathcal{E}=4.5$, $d=1$ (red solid line), $d=3$ (blue dot-dashed line), $d=6$ (green dashed line), and $d=8$ (yellow dotted line). In panel (b) the parameters are $d=2.2$, $\mathcal{E}=1.6$ (red solid line), $\mathcal{E}=2.5$ (blue dotted line), and $\mathcal{E}=5$ (green dot-dashed line).

Figure 7

(a) Sketch of possible semiclassical trajectories of particles entering the barrier region at different incident angles, as explained in the text. Panels (b) and (c): transmission probability as a function of the normalized width of the barrier $d/d_{\max }$ for the ${\mathcal{T}}_3$ lattice (solid red lines) and HCL (dashed blue lines). Panel (b): $\mathcal{E}=6.9$ and $\varphi =-\pi /4$. Panel (c): $\mathcal{E}=3.9$ and $\varphi =0$.