Tunneling spectroscopy between one-dimensional helical conductors

We theoretically investigate the tunneling spectroscopy of a system of two parallel one-dimensional helical conductors in the interacting, Luttinger liquid regime. We calculate the nonlinear differential conductance as a function of the voltage bias between the conductors and the orbital momentum shift induced on tunneling electrons by an orthogonal magnetic field. We show that the conductance map exhibits an interference pattern which is characteristic to the interacting helical liquid. This can be contrasted with the different interference pattern from tunneling between regular Luttinger liquids which is governed by the spin-charge separation of the elementary collective excitations.

Figure 1

Sketch of two possible scenarios of tunneling between two quantum wires (left) or edge states of a bilayer system of a topological insulator (right). The 1D conducting modes (shaded areas) are helical, i.e., opposite spin directions are bound to opposite motion as indicated by the arrows. While the spin directions in a bilayer system are generally parallel, they can differ between individual quantum wires, represented by the rotated symbols ${\uparrow }_2{,\downarrow }_2$ on the left. Tunneling between the conductors (wiggled lines) is of amplitude $\lambda$ and takes place over the length $\mathcal{L}$ of the upper conductor. A magnetic field $\mathbf{B}$ is applied perpendicular to the tunneling plane and provides an orbital momentum shift that allows compensation of the momentum mismatch between the Fermi points of the upper and lower conductors (see Fig. 2), and a voltage $V$ is applied between the conductors.

Figure 2

Possible tunneling processes. The upper panels show the band structure of conductor $n=1$ and the lower panel of conductor $n=2$. The colors and arrows indicate the spin projections of the helical bands. The tunneling is spin conserving and the magnetic field must be tuned such that the momentum $q_B$ compensates for the mismatch of alignment of bands of the same spin projection. Case (a) corresponds to bands with identical spin structure but different densities (e.g., by different electrostatic environments or gating). Case (b) represents the special case of exactly opposite spin structures, requiring a large $q_B$ value going across both Fermi momenta. In the general case of noncollinear spin directions between the conductors both cases (a) and (b) are possible, weighted by the corresponding spin overlap matrix elements ${\chi }_{{\sigma }_1,{\sigma }_2^{\prime }}=\langle {\sigma }_1|{\sigma }_2^{\prime }\rangle$.

Figure 3

Conductance map for tunneling between HLLs as a function of voltage $V$ and magnetic field $B$ (through $q_B=eBd$). Assumed is ${\sigma }_1={\sigma }_2$ such that tunneling is only possible between the modes $R\to R$ and $L\to L$. The used parameters for all plots are $\mathcal{L}=6\mu \mathrm{m}$, $v_F=2\times {10}^5$ m/s, $k_{F1}={10}^8{\mathrm{m}}^{-1}$, and $k_{F2}=1.02k_{F1}$ (at $V=0$), $a=5\AA ,\lambda =1\mathrm{meV},\beta =8$, and interaction parameters $K_1=0.8$, $K_2=0.6$. Since we set $v_1=v_2=v$ we choose $v=2v_F/(K_1+K_2)$ but notice that the precise interference beating pattern depends on this choice. The gray area marks $G=0$.

Figure 4

Conductance part $G_{\mathcal{L}}$ as in Fig. 3, multiplied by ${(eVL/2\pi v_F)}^{2-\gamma }$ to enhance the interference effect by suppressing the power-law dependence on $V$. The interference pattern is the superposition of the sinusoidal and Bessel function oscillations of each tunneling process.

Figure 5

Conductance part showing the interference pattern for tunneling between regular Luttinger liquids, as taken from Ref. [23], for the same parameters as in Fig. 3, taking $K_{1,2}$ as the charge interaction strengths. The corresponding spin parameters are set to $K=1$ [representing a preserved spin SU(2) symmetry]. The most notable difference to tunneling between HLLs are the vertical blurry interference fringes arising from the spin-charge separation. Notice that these fringes run continuously from the top to the bottom of the plot.

Figure 6

Illustration of the change of the interference patterns as a function of the interaction strengths $K_1$ and $K_2$. Shown is $G_{\mathcal{L}}{(eVL/2\pi v_F)}^{2-\gamma }$ as in Fig. 4. The calculation is only exact for the plots on the diagonal where $K_1=K_2$, and for $K_1\ne K_2$ we expect a further beating pattern from the velocity difference $v_1-v_2$. Notice that only in the noninteracting case $K_1=K_2=1$ the patterns exhibit horizontal uninterrupted beating fringes.