Topological transconductance quantization in a four-terminal Josephson junction

Recently we predicted that the Andreev bound-state spectrum of four-terminal Josephson junctions may possess topologically protected zero-energy Weyl singularities, which manifest themselves in a quantized transconductance in units of $4e^2/h$ when two of the terminals are voltage biased [R.-P. Riwar, M. Houzet, J. S. Meyer, and Y. V. Nazarov, Nature Commun. 7, 11167 (2016)]. Here, using the Landauer-Büttiker scattering theory, we compute numerically the currents flowing through such a structure in order to assess the conditions for observing this effect. We show that the voltage below which the transconductance becomes quantized is determined by the interplay of nonadiabatic transitions between Andreev bound states and inelastic relaxation processes. We demonstrate that the topological quantization of the transconductance can be observed at voltages of the order of ${10}^{-2}\mathrm{\Delta }/e,\mathrm{\Delta }$ being the the superconducting gap in the leads.

Figure 1

Sketch of a four-terminal Josephson junction. The normal-state scattering region connecting the four leads is described by the scattering matrix $\widehat{S}$. For the transconductance measurement, leads one and two are voltage biased with voltages $V_1$ and $V_2$, respectively. The other two leads, lead zero and lead three, are connected into a loop, such that the phase difference ${\varphi }_0-{\varphi }_3$ can be controlled by a magnetic flux through that loop. Gauge invariance allows us to choose ${\varphi }_3=0$. We compute the outgoing currents into all four leads.

Figure 2

Cuts through the Andreev bound-state spectrum for the scattering matrix ${\widehat{S}}_1$. From left to right: At phase ${\varphi }_0=1.72$ (Weyl point), at phase ${\varphi }_0=2.21$ where the gap is the largest within the topological region $\left(E=0.11\mathrm{\Delta }\right)$, and at phase ${\varphi }_0=2.66$ (Weyl point). The spectra are along the lines $({\varphi }_1=\chi ,{\varphi }_2=3\chi +\varphi )$, where the phase shift $\varphi$ is chosen such that the cut goes through the point in the $({\varphi }_1,{\varphi }_2)$ plane with the smallest gap.

Figure 3

The currents $I_0,I_1,I_2,I_3$ as function of voltage for the scattering matrix ${\widehat{S}}_1$. The voltages in terminals one and two are given as $V_1=2V$ and $V_2=3V$, respectively. The Dynes parameter is set to $\mathrm{\Gamma }=0.002\mathrm{\Delta }$. Top: At phase ${\varphi }_0=2.21$ in the topological region. Bottom: At phase ${\varphi }_0=0$ in the trivial region. We have used an average over $N=10$ phase shifts $\varphi$. The insets show a larger range of voltages.

Figure 4

The conductances $G_{12},G_{21},G_{11}$, and $G_{22}$ between the voltage-biased leads one and two as a function of voltage in logarithmic scale. The conductances are obtained from the currents shown in Fig. 3 and a similar set is obtained with a different voltage ratio, $V_1=V$ and $V_2=3V$. Top: At phase ${\varphi }_0=2.21$ in the topological region. Bottom: At phase ${\varphi }_0=0$ in the trivial region. The expected quantization of the transconductance is seen for voltages $eV/\mathrm{\Delta }\lesssim 0.01$.

Figure 5

Conductances as a function of phase ${\varphi }_0$ at fixed voltage. Top: $V=0.005\mathrm{\Delta }/e$. Bottom: $V=0.003\mathrm{\Delta }/e$. The quantized conductance plateaus are clearly visible. Around the topological transitions at $\pm {\varphi }_0^{\left(1\right)}=\pm 1.72$ and $\pm {\varphi }_0^{\left(2\right)}=\pm 2.66$, conductance quantization breaks down because the gap closes and dissipation becomes large.

Figure 6

Dissipation $P/P_N$ (dashed lines) and chirality $C$ (solid lines) as a function of voltage in logarithmic scale. Green curves correspond to $V_1=2V,V_2=3V$ and red curves correspond to $V_1=V,V_2=3V$. Top: At phase ${\varphi }_0=2.21$ in the topological region. The chirality tends to 1 as the dissipation tends to zero at low enough voltages. Bottom: At phase ${\varphi }_0=0$ in the trivial region.

Figure 7

To observe the quantization of the transconductance, it is favorable to have $({\varphi }_1,{\varphi }_2)$ planes with a large spectral gap both in the topological and the trivial region. Here we show a histogram of the smaller of the maximal gaps in these two regions for 500 random scattering matrices with Weyl points.

Figure 8

Dependence of the currents on the phase shift $\varphi$. Here ${\varphi }_0=2.21,\mathrm{\Gamma }=0.002\mathrm{\Delta }$, and $V=0.005\mathrm{\Delta }/e$. Left: ${\varphi }_1=eVt,{\varphi }_2=3eVt+\varphi$. Right: ${\varphi }_1=2eVt,{\varphi }_2=3eVt+\varphi$. We see that already 10 phase shifts give a rather good sampling.

Figure 9

Dissipation and chirality as a function of phase ${\varphi }_0$ at fixed voltage $V=0.003\mathrm{\Delta }/e$. The two curves for dissipation correspond to two sets of voltages $V_1=n_{1}V,V_2=n_{2}V$.

Figure 10

The currents $I_0,I_1,I_2,I_3$ as function of voltage for the scattering matrix ${\widehat{S}}_2$. The voltages in terminals one and two are given as $V_1=n_{1}V$ and $V_2=n_{2}V$, respectively (top, $n_1=1,n_2=3$; bottom, $n_1=2,n_2=3)$. The Dynes parameter is set to $\mathrm{\Gamma }=0.0001\mathrm{\Delta }$. Left: At phase ${\varphi }_0=2.16$ in the topological region. Right: At phase ${\varphi }_0=0$ in the trivial region. We have used an average over $N=10$ phase shifts $\varphi$. The insets show a larger range of voltages.

Figure 11

The conductances $G_{12},G_{21},G_{11}$, and $G_{22}$ between the voltage-biased leads one and two as a function of voltage in logarithmic scale. The conductances are obtained from the currents shown in Fig. 10. Left: At phase ${\varphi }_0=2.16$ in the topological region. Right: At phase ${\varphi }_0=0$ in the trivial region. The expected quantization of the transconductance is seen for voltages $eV/\mathrm{\Delta }\lesssim 0.001$.

Figure 12

Dissipation $P/P_N$ (dashed lines) and chirality $C$ (solid lines) as a function of voltage in logarithmic scale. Green curves correspond to $V_1=2V,V_2=3V$ and red curves correspond to $V_1=V,V_2=3V$. Left: At phase ${\varphi }_0=2.16$ in the topological region. The chirality tends to $-1$ as the dissipation tends to zero at low enough voltages. Right: At phase ${\varphi }_0=0$ in the trivial region.