Majorana STM as a perfect detector of odd-frequency superconductivity

We propose a novel scanning tunneling microscope (STM) device in which the tunneling tip is formed by a Majorana bound state (MBS). This peculiar bound state emerges at the boundary of a one-dimensional topological superconductor. Since the MBS has to be effectively spinless and local, we argue that it is the smallest unit that shows itself the properties of odd-frequency superconducting pairing. Odd-frequency superconductivity is characterized by an anomalous Green's function, which is an odd function of the time arguments of the two electrons building the Cooper pair. Interestingly, our Majorana STM can be used as the perfect detector of odd-frequency superconductivity. The reason is that a supercurrent between the Majorana STM and any other superconductor can only flow if the latter system exhibits itself odd-frequency pairing. To illustrate our general idea, we consider the tunneling problem of the Majorana STM coupled to a quantum dot placed on a surface of a conventional superconductor.

Figure 1

Majorana STM. The scanning tip contains the MBS described by $\gamma$, which is used to probe—via tunneling coupling ${\mathrm{t}}_{\sigma }$—either an unknown superconductor (SC) with odd-frequency pairing or a quantum dot (QD), which can realize odd-frequency pairing by proximity to an s-wave superconductor and an external magnetic field.

Figure 2

The critical current $I_c$ (solid lines) in comparison with the odd-frequency order parameter ${\partial }_t\left|{\mathbf{F}}_t^c\left(0\right)\right|$ (dashed lines) and the even-frequency order parameter $|F_s^c\left(0\right)|$ (dashed-dotted lines) for $B_z=0$, $U={\mathrm{\Gamma }}_{\mathrm{\Delta }}$, weak coupling $\mathrm{t}=0.1{\mathrm{\Gamma }}_{\mathrm{\Delta }}$, and temperature $T=0.2{\mathrm{\Gamma }}_{\mathrm{\Delta }}$.

Figure 3

Critical current of the SC-QD-MBS system (solid lines) as a function of the noncollinear component of the external magnetic field $B_{\perp }$ compared to the even-frequency singlet $|F_0|$ (dashed-dotted lines) and the magnitude of the odd-frequency triplet $|{\mathbf{F}}_t|$ (dashed lines) of the induced pairing in the dot. The current is calculated using Eq. (20) with $\theta =0$, $B_z=0$, $t=3\mathrm{\Delta }/2$, $t_{\uparrow }=0.1\mathrm{\Delta }$, $\beta =1/\left(k_{B}T\right)=5\mathrm{\Delta }$, and $\mathrm{\Delta }=1$. The pairing amplitudes are normalized by the scaling factor $(t_{\uparrow }^2/t^2)$. This figure nicely confirms the results displayed in Fig. 2 on the basis of Green's functions calculation.