Tunnel junction of helical edge states: Determining and controlling spin-preserving and spin-flipping processes through transconductance

When a constriction is realized in a 2D quantum spin Hall system, electron tunneling between helical edge states occurs via two types of channels allowed by time-reversal symmetry, namely spin-preserving (p) and spin-flipping (f) tunneling processes. Determining and controlling the effects of these two channels is crucial to the application of helical edge states in spintronics. We show that, despite that the Hamiltonian terms describing these two processes do not commute, the scattering matrix entries of the related 4-terminal setup always factorize into products of p-term and f-term contributions. Such factorization provides an operative way to determine the transmission coefficients $T_p$ and $T_f$ related to each of the two processes, via transconductance measurements. Furthermore, these transmission coefficients are also found to be controlled independently by a suitable combination of two gate voltages applied across the junction. This result holds for an arbitrary profile of the tunneling amplitudes, including disorder in the tunnel region, enabling us to discuss the effect of the finite length of the tunnel junction, and the space modulation of both magnitude and phase of the tunneling amplitudes.

Figure 1

A tunnel junction coupling the helical edge states can be realized by etching a constriction in a four-terminal quantum spin Hall effect setup. Two side gates enable one to tune the potential along the junction. Two types of time-reversal symmetric tunnel processes occur at the junction, namely spin-preserving and spin-flipping processes. Determining and controlling these two processes is crucial for exploiting helical edge states in spintronics devices.

Figure 2

The operative way to extract the transmission coefficients $T_p$ and $T_f$ related to spin-preserving and spin-flipping tunneling, respectively. When a voltage bias $V_2$ is applied to terminal 2, the currents flowing in the other three terminals directly depend on $T_p$ and $T_f$ in a factorized form, so that $T_p$ and $T_f$ can be determined via Eqs. (46) and (47). Furthermore, $T_p$ is controlled by the charge gate $V_{gc}=(V_{g,T}+V_{g,B})/2$, whereas $T_f$ is controlled by the spin gate $V_{gs}=(V_{g,T}-V_{g,B})/2$.

Figure 3

Effects of the spin-preserving and spin-flipping tunneling processes on the electronic spectrum of the edge states in the tunnel junction with profile (50) and (51). (a) The spin-preserving tunneling ${\Gamma }_p$ tends to induce a gap in the spectrum (which in fact amounts to a reduction of the transmission, due to the finite length of the junction), whereas the charge bias $V_{gc}$ shifts vertically the spectrum with respect to the Fermi level. (b) The spin-flipping tunneling ${\Gamma }_f$ lifts the degeneracy of spin-$\uparrow$ and spin-$\downarrow$ edge modes, inducing a mixing of the eigenstate spin components in the tunnel junction as well as a mutual horizontal shift in the momenta. The spin gate $V_{gs}$ renormalizes such shift. (c) The effect of both tunneling terms.

Figure 4

The case of a tunnel junction with a constant profile of the tunneling amplitude (50). (a) The transmission coefficient $T_p$ due to spin-preserving tunneling (evaluated at the equilibrium Fermi level $E_F$) is plotted as a function of $E_F-e{V}_{gc}$ (with $V_{gc}$ denoting the charge gate voltage applied in the tunnel region), for different values of the length of the junction. In the energy range $|E_F-e{V}_{gc}|<|{\Gamma }_p|$, $T_p$ exhibits a minimum, which gradually becomes a “gap” with increasing length of the junction [see also Fig. 2], whereas in the energy range $|E_F-e{V}_{gc}|>|{\Gamma }_p|$ one can observe oscillations due to the interference of electronic waves backward scattered at the ends of the tunnel junction. The amplitude and frequency of the oscillations depend on the spin-preserving strength of the tunnel junction $a_p=\left|{\Gamma }_p\right|L/\hslash v_F$: the oscillations increase in depth and frequency with increasing $a_p$. (b) The transmission coefficient $T_f$ due to spin-flipping tunneling is plotted as a function of the spin gate voltage $V_{gs}$ for various values of the length of the junction. In this case no “gapped” energy range is observed [see also Fig. 2]. The oscillations are due to forward scattering interference and increase in amplitude and frequency with increasing $a_f=\left|{\Gamma }_f\right|L/\hslash v_F$.

Figure 5

The behavior of the transmission coefficients $T_p$ and $T_f$ as a function of the length $L$ of the tunnel junction is typically nonmonotonic. (a) The spin-preserving transmission coefficient $T_p$ (evaluated at the energy $E_F=0$) is plotted as a function of $L|{\Gamma }_p|/\hslash v_F$, for various charge gate voltages. While for $|e{V}_{g,c}|>|{\Gamma }_p|$ it oscillates with $L$, in the regime $|e{V}_{g,c}|<|{\Gamma }_p|$ it exponentially decreases. (b) $T_f$ is plotted as a function of the length of the junction for various voltages $V_{gs}$. The behavior is always oscillatory. For particular values of the junction length $L|{\Gamma }_f|/\hslash v_F=(m+1/2)\pi$ (at $e{V}_{gs}=0$) the coefficient $T_f$ vanishes, leading to a complete spin flip.

Figure 6

Sketch of the tunneling amplitude profile ${\Gamma }_{\nu }\left(x\right)$ in the tunnel junction region $x_1\le x\le x_2$ ($\nu =p$ for spin-preserving and $\nu =f$ for spin-flipping tunneling processes, respectively). An arbitrary profile can be approximated with a staircase profile, and the transfer matrix is easily obtained as a product of the constant profile transfer matrices [see Eq. (58)].

Figure 7

(a) Sketch of a tunnel amplitude profile ${\Gamma }_{\nu }\left(x\right)$, where the transition from a vanishing tunneling coupling to a bulk value ${\Gamma }_{\nu }^0$ occurs over a smoothing length scale $\lambda$ (here $\nu =p,f$ for spin-preserving and spin-flipping processes, respectively), where $L$ is the total length of the junction. (b) The effect of a smoothing length $\lambda$ on the spin-preserving transmission coefficient: $T_p$ (evaluated at the equilibrium Fermi level $E_F$) is plotted as a function of $E_F-e{V}_{gc}$, for different values of the ratio $\lambda /L$, for the case $L|{\Gamma }_p^0|/\hslash v_F=5$. While the minigap region is essentially unaffected by $\lambda$, the amplitude and frequency of the oscillations are reduced as $\lambda /L$ increases. They are still visible as long as one can define a longitudinal “bulk” of the tunnel junction with a constant value ${\Gamma }_0$, i.e., for $\lambda /L<0.25$. (c) The effect of a smoothing length $\lambda$ on the spin-flipping transmission coefficient: $T_f$ is plotted as a function of $e{V}_{g,s}$, for the case $L|{\Gamma }_f^0|/\hslash v_F=5$. With increasing $\lambda$, the minimum at $V_{gs}=0$ exhibits a nonmonotonic behavior, decreasing for $\lambda /L=0.1$, and then increasing and even turning into a local maximum for $\lambda /L=0.3$. Similarly to the oscillations of $T_p$, also for $T_f$ the amplitude of the oscillations is suppressed when the tunnel junction profile becomes very smooth.

Figure 8

(a) The phase ${\varphi }_{\nu }$ of the tunneling amplitude ${\Gamma }_{\nu }\left(x\right)=\left|{\Gamma }_{\nu }\left(x\right)\right|exp\left[i{\varphi }_{\nu }\left(x\right)\right]$ is assumed to fluctuate inside the junction over a typical length scale ${\lambda }_{{\varphi }_{\nu }}$, whereas the absolute value $|{\Gamma }_{\nu }\left(x\right)|$ is assumed to be constant ($\nu =p$ for spin-preserving and $\nu =f$ for spin-flipping tunneling). (b) The spin-preserving transmission coefficient $T_p$ (evaluated at the equilibrium Fermi energy $E_F$) as a function of $E_F-e{V}_{gc}$, for the case $L|{\Gamma }_p|/\hslash v_F=5$. With respect to the case of constant ${\varphi }_p$ (black thin line), the fluctuations of ${\varphi }_p$ induce resonance maxima which appear inside the the “gap” region $|E_F-e{V}_{gc}|<|{\Gamma }_p|$, which in turn also broadens, whereas the oscillations in the “supragap” region $|E_F-e{V}_{gc}|>|{\Gamma }_p|$ are enhanced (blue thick line). The curve becomes also asymmetric with respect to $E_F-e{V}_{gc}$. (c) The spin-flipping transmission coefficient $T_f$ is plotted as a function of $e{V}_{g,c}$, for the case $L|{\Gamma }_f|/\hslash v_F=1.5$: the fluctuations of the phase ${\varphi }_f$ modify the oscillatory pattern with respect to the case of constant ${\varphi }_f$ (black thin line), by increasing the amplitude and making the plot asymmetric with respect to $V_{gs}=0$ (blue thick line).