Interfacial charge and spin transport in ${\mathbb{Z}}_2$ topological insulators

The Kane-Mele model realizes a two-dimensional version of a ${\mathbb{Z}}_2$ topological insulator as an idealized model of graphene with intrinsic and extrinsic (Rashba) spin-orbit couplings. We study the transport of charge and spin in such a Dirac electron system in the presence of a sharp potential step, that is, a $\mathit{pn}$ junction. An electron incident normal to the junction is completely reflected when Rashba coupling is dominant, whereas it is perfectly transmitted when the two types of couplings are balanced. The latter manifests in charge transport as a peak of conductance and a dip in Fano factor. Charge transport occurs in the direction normal to the barrier, whereas a spin current is induced along the barrier that is also localized in its vicinity. It is demonstrated that contributions from interband matrix elements and evanescent modes are responsible for such an interfacial spin Hall current. Our analysis of spin transport is based on the observation that in the case of vanishing Rashba coupling, each channel carries a conserved spin current, whereas only the integrated spin current is a conserved quantity in the general case. The perfect transmission/reflection of charge and conserved spin current is a consequence of reflection symmetry. Finally, we provide a quasiclassical picture of charge and spin transport by imaging flow lines over the entire sample and Veselago lensing (negative refraction).

Figure 1

Potential step $V(x)$.

Figure 2

Top: Transmission probability $T(p_y=0)$ at the normal incidence as a function of $V_0/\Delta$ and $E/\Delta$ at ${\lambda }_R/\Delta =1.2$. Bottom: Dependence of $T(p_y=0)$ on ${\lambda }_R/\Delta$ for a fixed value of $(E/\Delta ,V_0/\Delta )=(0.5,2)$. In the upper panel, each region corresponds to a finite range of $T(p_y=0)$ indicated. In the lower panel, each curve corresponds to different values of $V_0/\Delta$.

Figure 3

Top: Charge conductance $G_c$ of the potential step (height $V_0$) normalized by $G_0={\mathit{Wp}}_F/\pi$ ($E/\Delta =0.5$). Bottom: Fano factor $F$. Red, blue, and green curves represent the typical behavior of charge conductance in the semimetallic phase (${\lambda }_R/\Delta =1.7$), in the topological gap phase (${\lambda }_R/\Delta =0.6$), and at the phase boundary (${\lambda }_R/\Delta =1$), respectively.

Figure 4

Charge conductance (top) vs. Fano factor (bottom) as a function of Rashba spin-orbit coupling. Incident energy is set to be $E/\Delta =0.5$. Each curve corresponds to different values of $V_0/\Delta$.

Figure 5

Spatial distribution of spin current density. ${\lambda }_R/\Delta =2$ (top) and ${\lambda }_R/\Delta =0.5$ (bottom). The spin Hall current is localized in the vicinity of the junction. $j_0=\Delta /(2\pi L)$.

Figure 6

Spin transport in the Kane-Mele $p$-$n$ junction for different values of ${\lambda }_R$. The $p$-$n$ junction induces an $s_z$-spin current in the direction parallel to the interface. Such a spin Hall current is plotted at $x=0$ as a function of $V_0$.

Figure 7

Mesoscopic spin Hall effect in the absence of a topological mass term, $\Delta =0$ (top), and of Rashba spin-orbit coupling, ${\lambda }_R=0$ (bottom). Spin Hall currents at $x=0$ are plotted as a function of $V_0$ for $E/{\lambda }_R=0.5$ with $\Delta =0$, $j_0={\lambda }_R/(2\pi L)$, and $E/\Delta =1.5$ with ${\lambda }_R=0$, $j_0=\Delta /(2\pi L)$.

Figure 8

Refraction of electron beams by the potential step in the topological gap phase (${\lambda }_R/\Delta =0.5$). (a) Intraband tunneling ($V_0/\Delta =-2$). (b) Interband tunneling ($V_0/\Delta =2$). Spatial coordinates: $\mathrm{x̃}=x\Delta /(\hslash v_F)$, $\mathrm{ỹ}=y\Delta /(\hslash v_F)$. Fermi energy: $E/\Delta =0.5$.

Figure 9

Electron Veselago lens: semimetallic phase (${\lambda }_R/\Delta =2$). Spatial coordinates: $\mathrm{x̃}=x\Delta /(\hslash v_F)$, $\mathrm{ỹ}=y\Delta /(\hslash v_F)$. Fermi energy: $E/\Delta =0.5$. The height of the potential step $V_0$ is chosen such that (a) $V_0/\Delta =2$ and (b) $V_0/\Delta =3.5$, both corresponding to the interband tunneling case. Case (b) corresponds to the opening of the lowest energy channel.

Figure 10

Detailed plots of the shaded region in Fig. 9b: ${\lambda }_R/\Delta =2$, $V_0/\Delta =3.5$. Other parameters are also the same.

Figure 11

Contribution of the $|-\rangle$ band to charge conductance. Top: Charge conductance $G_{-}$ calculated from $j_{-}$. Bottom: Charge conductance $G_{+}$ calculated from $j_{+}$. The system’s parameters are chosen such that ${\lambda }_R/\Delta =1.7$, $E/\Delta =0.5$.