Effects of disorder on electron tunneling through helical edge states

A tunnel junction between helical edge states, realized via a constriction in a quantum spin Hall system, can be exploited to steer both charge and spin current into various terminals. We investigate the effects of disorder on the transmission coefficient $T_p$ of the junction by modeling disorder with a randomly varying (complex) tunneling amplitude ${\Gamma }_p=\left|{\Gamma }_p\right|exp\left[i{\varphi }_p\right]$. We show that, while for a clean junction $T_p$ is only determined by the absolute value $|{\Gamma }_p|$ and is independent of the phase ${\varphi }_p$, the situation can be quite different in the presence of disorder: phase fluctuations may dramatically affect the energy dependence of $T_p$ of any single sample. Furthermore, analyzing three different models for phase disorder (including correlated ones), we show that not only the amount but also the way the phase ${\varphi }_p$ fluctuates determines the localization length ${\xi }_{\mathrm{loc}}$ and the sample-averaged transmission. Finally, we discuss the physical conditions in which these three models suitably apply to realistic cases.

Figure 1

A constriction etched in a four-terminal quantum spin Hall effect setup induces an electron-tunnel coupling between the helical edge states. Vertical arrows inside the circles indicate electron spin orientation. A fluctuating width along the longitudinal direction $x$ causes geometrical disorder, while electrical disorder can originate from the presence of oxides (small red dots) produced by the etching process and/or from the potential fluctuations (bigger green dots) due to the amorphous dielectric separating the conducting channel from the top gate.

Figure 2

Transmission coefficient for a disordered tunnel junction where the (complex) tunneling amplitude ${\Gamma }_p=\left|{\Gamma }_p\right|exp\left[i{\varphi }_p\right]$ is randomly varying. The disorder on the absolute value $|{\Gamma }_p|$ (solid blue curve) has different effects than the disorder on the phase ${\varphi }_p$ (dashed red curve). For comparison, the thin black line denotes the clean case. Here $L/{\xi }_0=5$ and $N=10$, corresponding to the regime $l_p\lesssim {\xi }_0<L$.

Figure 3

Transmission coefficient for a disordered tunnel junction with $L/{\xi }_0=3$ and $N=500$, corresponding to the regime $l_p\ll {\xi }_0<L$. The absolute value $|{\Gamma }_p|$ varies according to the Rayleigh distribution, and the phase is uniformly picked in a range $[-\Delta {\varphi }_p/2;\Delta {\varphi }_p/2]$. While the fluctuations of the absolute value $|{\Gamma }_p|$ of the tunneling amplitude do not modify the clean-case result significantly, the amount $\Delta {\varphi }_p$ of phase fluctuation may have a dramatic impact. In particular, while for $\Delta {\varphi }_p<\pi$ minor modifications appear on $T_p$, when $\Delta {\varphi }_p\simeq 2\pi$ (corresponding to a Gaussian-distributed tunneling amplitude), the phase fluctuations wash out the tunneling term, so that the transmission coefficient becomes energy independent and tends to 1.

Figure 4

Three different models of disorder fluctuations for the phase ${\varphi }_p$ of the tunneling amplitude [see Eqs. (37, 38, 39)]. The horizontal axis represents the longitudinal coordinate along the junction, divided into $N$ intervals with size $l_p$.

Figure 5

The (inverse) localization length ${\xi }_{\mathrm{loc}}$, in units of ${\xi }_0$, plotted as a function of energy for the three different models of phase fluctuations depicted in Fig. 4 and described in the text. The quite different behaviors are discussed in the text. The disorder length scale is $l_p=2{\xi }_0$ for all models. For the c-RW model $\Delta {\varphi }_p=\pi /2$, and for the c-LC model the parameter is $\mathrm{SD}\left(k_p\right)=2/{\xi }_0$. The dotted black curve represents the clean-case result ${\xi }_0/{\xi }_{cl}\left(E\right)$ [see Eq. (36)].

Figure 6

Energy dependence of sample-averaged transmission: the average transmission $\overline{T_p}$ (black solid curve), the typical transmission $T_p^{\mathrm{typ}}$ [see Eq. (40); red dashed curve], and the transmission fluctuations $\delta T_p$ [see (41); blue dotted curve]. Here $l_p=2{\xi }_0$, $N=6$, and averaging has been performed over $6\times {10}^3$ samples. (a) The u-G model and (b) the c-LC model for phase fluctuations (see Fig. 4). In both cases the behavior of ${\xi }_0/{\xi }_{\mathrm{loc}}$ has also been reported (thin black solid curve) as a guide to the eye. The disorder parameter for the c-LC model is $\mathrm{SD}\left(k_p\right)=2/{\xi }_0$.

Figure 7

Various sources of disorder in a helical tunnel junction: (a) geometrical disorder and (b) and (c) electrical disorder. See text for details. For the geometrical disorder the fluctuations $\delta W$ of the width are typically smaller than the average width ($\delta W\ll W)$, so that $|{\Gamma }_p|\simeq \langle |{\Gamma }_p|\rangle$ is weakly disordered. However, the difference $\delta s^{\left(j\right)}=\delta s_1^{\left(j\right)}-\delta s_2^{\left(j\right)}$ between opposite profile lengths may cause fluctuations on the phase ${\varphi }_p$ of the tunneling amplitude. In (b) and (c) the shaded area represents the typical profile of electrical disorder along the longitudinal direction of the helical tunnel junction. (b) and (c) describe the cases of long- and short-range potential, respectively, compared to the typical potential center distance $l_p$. The red dashed curves represent the equivalent formulation in terms of the phase of the tunneling amplitude after applying the gauge transformation equation (12).