Transport Signatures of Fractional Fermions in Rashba Nanowires

We theoretically study transport through a semiconducting Rashba nanowire (NW) in the presence of uniform and spatially modulated magnetic fields. The system is fully gapped, and the interplay between the spin orbit interaction and the magnetic fields leads to fractionally charged fermion (FF) bound states of the Jackiw-Rebbi type at each end of the nanowire. We investigate the transport and noise behavior of a $\mathrm{N}/\mathrm{NW}/\mathrm{N}$ system, where the wire is contacted by two normal leads (N), and we look for possible signatures that could help in the experimental detection of such states. We find that the differential conductance and the shot noise exhibit a subgap structure which fully reveals the presence of the FF state. Alternatively, another confirmation of the presence of the FFs is provided by a conductance measurement in an Aharonov-Bohm setup, where the FFs are responsible for oscillations with double period. Our predictions can be tested in $\mathrm{InSb}/\mathrm{InAs}$ nanowires and are within reach of the present technology.

Figure 1

Sketch of the transport setup, consisting of a semiconducting NW (pink, light gray) of length $L$ attached to normal (N) leads (blue, black). The chemical potential in the NW is controlled by the gate (G) (light blue, light gray). A uniform magnetic field $\mathit{B}$ is applied along the wire. The NW is also subjected to a spatially varying magnetic field ${\mathit{B}}_n(x)$ produced by periodically arranged nanomagnets (green, light gray). A fractional-fermion bound state forms at each wire end (red, light gray).

Figure 2

Energies $E_{\text{FF}}(\theta )$ and $E_{\text{FF}}^{\prime }({\theta }^{\prime })$ of the left (full lines) and right (dashed lines) FFs bound state, for $L\to \infty$, for different gap values, as evaluated from the analytical result Eq. (3). The angle ${\theta }^{\prime }$ describes the right-end boundary condition for ${\mathit{B}}_n$ and is given by $\theta +4k_{\text{so}}L$. Here we chose symmetric boundary conditions ${\theta }^{\prime }=\theta +n\pi$, in which case the two FF energies vanish for $\theta =\pi$, merging into the continuum at $E=\min \{{\mathrm{\Delta }}_z,{\mathrm{\Delta }}_n\}$, in correspondence of $\overline{\theta }=\overline{\theta }({\mathrm{\Delta }}_z/{\mathrm{\Delta }}_n)$.

Figure 3

Tight-binding numerical results for a nanowire with a normal/fractional fermion ($\mathrm{N}/\mathrm{FF}$) junction, with FF bound states in the FF phase. ${\mathrm{\Delta }}_z=0.2t_x$ and ${\mathrm{\Delta }}_n=0.1t_x$ ($t_x$ hopping parameter). The rotating field $B_n$ is present only in the FF section of length $L_{\mathrm{T}}$, with $L_{\mathrm{T}}=3L_{\mathrm{N}}$ and $L_{\mathrm{N}}=100$. At the $\mathrm{N}/\mathrm{FF}$ interface there is a rectangular barrier of width $L_b=L_{\mathrm{T}}/50$ and height $V=5t_x$ (left column) or $V=t_x$ (right column). (a)—(b): TB energy spectrum $E_n(\theta )$. Inside the bulk gap, delimited by continuum states (black dots), there are two FF bound states, localized at the left (red dots) and right end (blue dots). In addition, there are states localized in the N section (cyan dots). (c)—(f): Probability densities $|{\psi }_{\sigma }(x){|}^2$ of the left FF state at $\theta =0.6\pi$ [(c),(d)] and $\theta =0.1\pi$ [(e),(f)]. If the barrier is high, the FF is completely localized in the FF section [(c),(d)], and as a result the FF spectrum shows no hybridization [(a)] and is fully consistent with Eq. (3). For lower barrier, the left FF leaks out in the normal section [(e),(f)] and gets hybridized with the N-region subgap states [(b)].

Figure 4

The coefficients $T_{\uparrow }=T_{\uparrow \uparrow }+T_{\uparrow \downarrow }$ [(a) and (c)] and $R_{\uparrow \downarrow }$ [(b) and (d)] are plotted in the $E$-$\theta$ plane, as evaluated in two different regimes. Panels (a) and (b) refer to the high-barrier (${\lambda }_1={\lambda }_2=10\alpha$), small-${\mu }_l$ limit ($k_l\simeq k_{\text{so}}/2$). Panels (c) and (d) are instead obtained in the zero-barrier, large-${\mu }_l$ limit ($k_l\simeq 100k_{\text{so}}$), which yields an equivalent degree of decoupling, but does not shift the bound state spectra, as highlighted by the white dotted lines, which correspond to Eq. (3). The other parameter values are ${\mathrm{\Delta }}_z={\epsilon }_{\text{so}}/10$, ${\mathrm{\Delta }}_n={\epsilon }_{\text{so}}/5$, $L\simeq 3{\xi }_z(0)$.

Figure 5

Aharonov-Bohm setup where the NW (pink, light gray) hosting a FF at each end connects the two arms of the AB ring (blue, black). When $\mu$ lies within the NW gap, only standard AB oscillations in the conductance $G(\mathrm{\Phi })$ as function of flux $\mathrm{\Phi }$ (generated by a perpendicular magnetic field $B_0$) appear, whereas when resonance with the degenerate FFs is reached, $G$ exhibits new oscillations with doubled period (see Fig. 6).

Figure 6

The dependence of the conductance $G(\nu ,V)$ (in arbitrary units) on frequency $\nu$ (conjugate to $\mathrm{\Phi }/{\mathrm{\Phi }}_0$) of the AB oscillations and applied voltage $V$ for the AB ring shown in Fig. 5. For $V$ within the gap of the wire (white dashed lines), the doubled periodicity (giving rise to additional conductance peaks at $\nu =1/2,3/2,\dots$) is observed only when $V$ matches the energy of the midgap FFs.