Floquet Majorana bound states in voltage-biased planar Josephson junctions

We study a planar Josephson junction under an applied DC voltage bias in the presence of an in-plane magnetic field. Upon tuning the bias voltage across the junction $V_{\mathrm{J}}$, the two ends of the junction are shown to simultaneously host both zero and $\pi$ Majorana modes. These modes can be probed using either a scanning-tunneling-microscopy measurement or through resonant Andreev tunneling from a lead coupled to the junction. While these modes are mostly bound to the junction's ends, they can hybridize with the bulk by absorbing or emitting photons. We analyze this process both numerically and analytically, demonstrating that it can become negligible under typical experimental conditions. Transport signatures of the zero and $\pi$ Majorana states are shown to be robust to moderate disorder.

Figure 1

(a) Proposed experimental setup. A voltage-biased planar Josephson junction implemented in a Rashba spin-orbit coupled 2DEG in the presence of an in plane-magnetic field. (b) Low-energy spectrum of the 2DEG strip in the absence of superconductivity. The Fermi level ($\varepsilon =0$) is marked by a solid black line, and $\varepsilon =\mathrm{\Omega }/2$ is marked by a gray dashed line, where three different scenarios are considered. A $\pi$ Majorana state emerges at the junction ends whenever $\mathrm{\Omega }/2$ crosses an odd number of bands. The scenarios (i) $V_{\text{J}}=43.75\mu \text{V}$, (ii) $V_{\text{J}}=125\mu \text{V}$, and (iii) $V_{\text{J}}=237.5\mu \text{V}$ correspond to Figs. 2, respectively. The parameters used to obtain the spectrum are the same as those given in the text, i.e., $E_{\mathrm{so}}=100\mu \mathrm{eV}$, $E_{\mathrm{Z}}^0=75\mu \mathrm{eV}$, etc. (see the end of Sec. 2), except that here the chemical potential is shifted to ${\mu }_{\mathrm{J}}=0\mu \mathrm{eV}$ in order to compensate for the absence of the superconductors.

Figure 2

Conductance as a function of the lead's voltage $V$ and the junction's length $L_x$, in units of $2e^2/h$. The voltage across the junction is taken to be (a) and (b) $V_{\text{J}}=43.75\mu \text{V}$, (c) and (d) $V_{\text{J}}=125\mu \text{V}$, and (e) and (f) $V_{\text{J}}=237.5\mu \text{V}$. These three values correspond to the three dashed gray lines in Fig. 1, marked (i), (ii), and (iii), respectively.

Figure 3

Conductance as a function of the lead's voltage $V$ and the junction's length $L_x$, in units of $2e^2/h$. System parameters are the same as in Figs. 2 and 2, except the coupling to the lead is reduced, enabling us to view the splitting of the Majorana resonance. The splitting does not decrease to zero with increasing $L_x$ and is most likely a result of photon-mediated coupling between Majorana states on opposite ends of the junction (see Sec. 4).

Figure 4

Conductance as a function of the lead's voltage $V$ for a fixed junction's length, $L_x=16\mu \mathrm{m}$, and different values of disorder strength represented by inverse mean-free time. System parameters are the same as in Figs. 2 and 2. In (a) we focus on voltages around $V=0$, and in (b) we focus on voltages around $V=V_{\mathrm{J}}$, corresponding to zero and $\pi$ Majorana states, respectively.

Figure 5

Conductance as a function of the lead's voltage $V$ and the junction's length $L_x$, in units of $2e^2/h$. The voltage across the junction is taken to be $V_{\text{J}}=125\mu \text{V}$.

Figure 6

Linecuts from Fig. 2, showing conductance versus voltage for a fixed system length of $L_x=16\mu \mathrm{m}$. Left: Zero Majorana. Right: $\pi$ Majorana.