Majorana-assisted nonlocal electron transport through a floating topological superconductor

The nonlocal nature of the fermionic mode spanned by a pair of Majorana bound states in a one-dimensional topological superconductor has inspired many proposals aiming at demonstrating this property in transport. In particular, transport through the mode from a lead attached to the left bound state to a lead attached to the right will result in current cross correlations. For ideal zero modes on a grounded superconductor, the cross correlations are however completely suppressed in favor of purely local Andreev reflection. In order to obtain a nonvanishing cross correlation, previous studies have required the presence of an additional global charging energy. Adding nonlocal terms in the form of a global charging energy to the Hamiltonian when testing the intrinsic nonlocality of the Majorana modes seems to be conceptually troublesome. Here, we show that a floating superconductor allows observing nonlocal current correlations in the absence of charging energy. We show that the noninteracting and the Coulomb-blockade regime have the same peak conductance $e^2/h$ but different shot-noise power; whereas the shot noise is sub-Poissonian in the Coulomb-blockade regime in the large-bias limit, Poissonian shot noise is generically obtained in the noninteracting case.

Figure 1

(a) Setup consisting of a one-dimensional topological superconductor (TS) that is floating and hosts ideal zero-energy Majorana bound states ${\gamma }_1,{\gamma }_2$ contacted by two normal-conducting noninteracting leads (N). The superconductor is subject to a charging energy $E_C$ such that a preferred number of charges is controllable through a gate voltage $V_g$. We study the system in the case $E_C=0$ and compare it to known results for large $E_C$. (b) In absence of charging energy $E_C=0$, the conductance and shot noise of the system can be understood in terms of an equivalent circuit consisting of (noiseless) conductances $G_i$ which are associated with local Andreev reflection off the bound state ${\gamma }_i$ and current sources generating noise currents $\delta I_i$.

Figure 2

Relation between the voltage drop $V_{min}$ at the terminal with the smaller lead coupling ${\mathrm{\Gamma }}_{min}$ and the voltage drop $V_{max}$ at the terminal with the larger lead coupling ${\mathrm{\Gamma }}_{max}$ for a floating superconductor.

Figure 3

Fano factor $F=S/eI$ in the floating system as a function of the voltage $V/V_{\mathrm{sat}}$, where $e{V}_{\mathrm{sat}}=2{\mathrm{\Gamma }}_{max}tan({\mathrm{\Gamma }}_{min}\pi /2{\mathrm{\Gamma }}_{max})$ is the largest voltage $|V_{max}|$ that can drop at the terminal with the larger lead coupling ${\mathrm{\Gamma }}_{max}$. For voltages $V\gg V_{\mathrm{sat}}$, nearly all the voltage drops at the terminal with the smaller lead coupling ${\mathrm{\Gamma }}_{min}$ and both the shot noise $S/e$ and the current $I$ tend to the maximal current $2\pi e{\mathrm{\Gamma }}_{min}/h$ that can be driven through the system. Consequently, for voltages $V\gg V_{\mathrm{sat}}$, the Fano factor reaches 1.

Figure 4

This plot shows the equivalence of the tunneling through a symmetric double barrier to the Andreev reflection off an NS interface. (a) The tunneling amplitudes for a double barrier $t={\sum }_{m=0}^{\infty }{A}_m$ is a sum of the amplitudes $A_m=t_L{e}^{i\phi }{\left(r_R{e}^{2i\phi }{r}_L\right)}^m{t}_R$ for $m$ round trips between the barriers with the amplitudes $r_{L/R}$ and $t_{L/R}$ of reflection and transmission at the left/right barriers and the phase $e^{i\phi }$ accumulated during one traversal of the barrier region. (b) The tunneling amplitude for Andreev reflection $t^{eh}$ at an NS interface is similarly given by the sum of the amplitude $A_m=t_e{e}^{i\chi }{\left(r_h{e}^{i\tilde{\chi }}{r}_h{e}^{i\chi }\right)}^m{t}_h$ for $m$ round trips with the amplitudes $r_{e/h}$ and $t_{e/h}$ for reflection and transmission of an electron/hole at the tunneling barrier and the amplitudes $e^{i\chi },e^{i\tilde{\chi }}$ for the Andreev reflection as a hole or electron at the superconductor. For energies much smaller than the gap, the particle-hole symmetry dictates that $\chi +\tilde{\chi }=-\pi$. Thus, we obtain the result [13] that Andreev reflection can be understood as transmission through a symmetric double barrier with $T_L=|t_L{|}^2=T_R={\left|t_R\right|}^2$ if we identify $\phi =-\pi /2$ and $t_{e/h}=t_{L/R},r_{e/h}=r_{L/R}$.