Super-Klein tunneling of massive pseudospin-one particles

The transmission properties of massive particles with pseudospin-one through homogeneous and heterogeneous junctions are studied from an effective spin-orbit Hamiltonian. The addition of a mass term in the Hamiltonian creates an energy band gap with a flat band inside the gap. There are three possible scenarios for the location of the flat band: at the top of the valence band, at the bottom of the conduction band, and at the center of the energy band gap. We have studied how the position of the flat band affects the transmission through a general type of junction. We found that omnidirectional perfect transmission, called super-Klein tunneling, occurs even for massive particles with specific symmetrical conditions in the junction. In all other cases, an angular independent transmission is obtained, which can be considered as an attenuated super-Klein tunnelling. These effects emerge when the junction operates as a Veselago lens under the generalized focusing condition. Furthermore, we found that Klein tunneling is restored in the massless limit. The present findings may have important implications in the development of electronic devices based on quantum optics with massive pseudospin-one particles.

Figure 1

Graphical representation of the dispersion relations of the Hamiltonian (1). (a) For the case of mass generation term $M=S_z$, while (b) and (c) correspond to the cases $M=U$ and $M=-U$, respectively.

Figure 2

(a) Scheme of a junction depicted by a step potential $V_0$ and the corresponding energy band structure. (b) Description of the particle scattering through the junction; a particle beam is injected from the left in region I, with an incidence angle $\varphi$. Then, a reflected beam comes back in region I, while a refracted beam is transmitted through region II with a refraction angle ${\varphi }^{\prime }$.

Figure 3

System $(0,0)$ with $M=S_z$ in both regions of the junction: (a) scheme of the band structure and (b) transmission probability as a function of the energy $E$ and the incidence angle $\varphi$, using the set of values $\mathrm{\Delta }={\mathrm{\Delta }}^{\prime }=40$ meV, $V_0=200$ meV, and the Fermi velocity $v_F=0.83\times {10}^6 {\mathrm{ms}}^{-1}$.

Figure 4

(a) Transmission probability as a function of $\varphi$ for the $(0,0)$ system: SKT behavior (red curve) is obtained with the set of values $V_0=200$ meV, $E=V_0/2=100$ meV, and $\mathrm{\Delta }={\mathrm{\Delta }}^{\prime }=40$ meV for equal masses in both sides of the junction. Orange (yellow) curve shows how SKT is lost when the value of energy is changed to $E=95$ ($E=90$) meV. Such an effect also is affected setting the masses to be different in both regions with $\mathrm{\Delta }=0$ ($\mathrm{\Delta }=90$) and ${\mathrm{\Delta }}^{\prime }=40$ meV, as shown in blue (green) curve, where $E=96$ ($E=116$) meV is calculated from the general focusing condition (13). (b) Transmission probability as a function of $\varphi$ for the $(1,1)$ system using the same set of values as in (a): red, blue, and green curves show ASKT (see text).

Figure 5

Scheme of the band structure for the systems (a) $(1,1)$ and (c) $(-1,-1)$. Transmission probability as a function of the incidence angle $\varphi$ and the energy $E$ for the systems (b) $(1,1)$ and (d) $(-1,-1)$, using the set of values $\mathrm{\Delta }={\mathrm{\Delta }}^{\prime }=40$ meV, $V_0=200$ meV, and the Fermi velocity value $v_F=0.83\times {10}^6 {\mathrm{ms}}^{-1}$.

Figure 6

Transmission probability as a function of the energy $E$ and the incidence angle $\varphi$, for the systems (a) $(1,-1)$ and (b) $(-1,1)$, with the same set of parameter values as in Fig. 5.

Figure 7

Transmission probability as a function of the energy $E$ and the incidence angle $\varphi$, for the systems (a) $(0,1)$, (b) $(1,0)$, (c) $(0,-1)$, and (d) $(-1,0)$.

Figure 8

Transmission probability for massive particles as a function of $E$ for normal incidence ($\varphi =0$), for the case of pseudospin-$1/2$ in green, and for the case of pseudospin-one in the configurations: $(1,-1)$ and $(-1,1)$ in red and $(0,0)$ in yellow.