Conductance quantization in topological Josephson trijunctions

Majorana zero modes are attracting a lot of attention as a new kind of emergent quasiparticles in condensed-matter physics. Here we propose a new and striking signature of the coupling between Majorana zero modes in a three-terminal Josephson junction made with topological superconductors. Recently three- and four-terminal junctions made either with conventional or with topological superconductors have been shown to exhibit Weyl crossings in their Andreev spectrum that lead to a nonzero quantized transconductance, like in the quantum Hall effect. In contrast with the other types of junctions, where the effect is limited to certain parameter regimes, we predict that, generically, any three-terminal Josephson junction made with topological superconductors is characterized by such a nonzero quantized transconductance. The quantization value is $\pm 2e^2/h$. Based on numerics, we check that this prediction is robust.

Figure 1

Sketch of a trijunction made of three topological superconductors hosting Majorana modes at their ends. (a) The superconducting phase difference ${\varphi }_2$ between two of the terminals can be tuned by a magnetic flux ${\mathrm{\Phi }}_{\text{ext}}$ applied through the superconducting loop that connects them. A voltage $V_1$ is applied to the remaining terminal. The transconductance $G_{21}=\partial I_2/\partial V_1$ is predicted to be quantized in units of $2e^2/h$ after averaging over ${\varphi }_2$. (b) Alternatively, one can measure the transconductance between two voltage-biased terminals.

Figure 2

Value of the gap $E_m={min}_{{\varphi }_1,{\varphi }_2}{E}_{+}$ as a function of the “chirality” $\delta G=G_{ii+1}^N-G_{i+1i}^N$ for 5000 random scattering matrices drawn from a CUE. It illustrates the correlation between these two quantities, $E_m/\mathrm{\Delta }\ge (3\sqrt{3}/8)\left|\delta G\right|/(e^2/h)$ (indicated by the black line).

Figure 3

Differential conductance $G_{ij}=\partial I_i/\partial V_j$ as a function of voltage (log-scale) with a Dynes parameter $\gamma =2\times {10}^{-3}\mathrm{\Delta }$ for a scattering matrix drawn from CUE with ${min}_{{\varphi }_1,{\varphi }_2}{E}_{+}\approx 0.37\mathrm{\Delta }$, $\delta G\approx -0.26e^2/h$. The transconductance is quantized in an intermediate voltage regime. At very low voltages, the flat band induces dissipation, whereas, at high voltages, multiple Andreev reflections [35, 36] produce an intricate pattern of subgap features. Finally, at voltages $eV\gg \mathrm{\Delta }$, the normal-state conductances are recovered.

Figure 4

Conductance as a function of flux $\mathrm{\Phi }$ through the junction area of a symmetric trijunction (see the inset for a sketch of its central region) with $U=0$, $t=1/\pi \nu$, and a Dynes parameter $\gamma ={10}^{-3}\mathrm{\Delta }$ at voltage $eV=0.1\mathrm{\Delta }$. Plateaus of quantized transconductance are visible away from time-reversal invariant values of the flux, $\mathrm{\Phi }/{\mathrm{\Phi }}_0=0,1/2$, while a large dissipation is generated in their vicinity.

Figure 5

Differential conductance $G_{ij}=\partial I_i/\partial V_j$ as a function of voltage for various random scattering matrices with a Dynes parameter $\gamma ={10}^{-3}\mathrm{\Delta }$. (a) ${min}_{{\varphi }_i}{E}_{+}\approx 0.35\mathrm{\Delta }$, $\delta G\approx 0.46e^2/h$. (b) ${min}_{{\varphi }_i}{E}_{+}\approx 0.40\mathrm{\Delta }$, $\delta G\approx -0.34e^2/h$. (c) ${min}_{{\varphi }_i}{E}_{+}\approx 0.39\mathrm{\Delta }$, $\delta G\approx 0.50e^2/h$.

Figure 6

Differential conductance $G_{ij}=\partial I_i/\partial V_j$ for various values of the Dynes parameter $\gamma =0.001$ (full lines), $\gamma =0.01$ (dashed lines), and $\gamma =0.05$ (dotted lines). The larger $\gamma$, the larger the voltage at which deviations from conductance quantization appear in the low-voltage regime.

Figure 7

Temperature dependence of the conductance $G_{ij}=I_i/V_j$ at voltage $V=0.1\mathrm{\Delta }/e$, and with a Dynes parameter (a) $\gamma ={10}^{-3}\mathrm{\Delta }$, (b) $\gamma ={10}^{-2}\mathrm{\Delta }$, and (c) $\gamma =5\times {10}^{-2}\mathrm{\Delta }$.

Figure 8

Conductance $G_{ij}=I_i/V_j$ as a function of the coupling $t_0$ at voltage $V=0.15\mathrm{\Delta }$ and with a Dynes parameter $\gamma ={10}^{-3}\mathrm{\Delta }$. Here $t=0.6$ and $U=0$.

Figure 9

Conductance $G_{ij}=I_i/V_j$ as a function of the coupling $\alpha$ at voltage $V=0.15\mathrm{\Delta }$ and with a Dynes parameter $\gamma ={10}^{-3}\mathrm{\Delta }$. Here $t=0.6$ and $U=0$.