Spin-dependent zero-bias peak in a hybrid nanowire-quantum dot system: Distinguishing isolated Majorana fermions from Andreev bound states

A hybrid system composed by a semiconducting nanowire with proximity-induced superconductivity and a quantum dot at the end working as a spectrometer was recently used to quantify the so-called degree of Majorana nonlocality [M.-T. Deng et al. Phys. Rev. B 98, 085125 (2018)]. Here we demonstrate that the spin-resolved density of states of the dot responsible for the zero-bias conductance peak strongly depends on the separation between the Majorana bound states and their relative couplings with the dot and investigate how the charging energy affects the spectrum of the system in the distinct scenarios of Majorana nonlocality (topological quality). Our findings suggest that the spin-resolved spectroscopy of the local density of states of the dot can be used as a powerful tool for discriminating between different scenarios of the emergence of the zero-bias conductance peak.

Figure 1

Sketch of the system consisting of a hybrid superconducting (SC) nanowire (blue region) coupled to a quantum dot (QD) with energy level ${\varepsilon }_d$, which can be tuned by application of an external gate voltage $V_{\text{Dot}}$. The QD is coupled to both Majorana bound states (MBSs) ${\gamma }_L$ and ${\gamma }_R$ at opposite ends of the SC nanowire with strengths ${\lambda }_L$ and ${\lambda }_R$, respectively. The MBSs may be hybridized with each other by ${\delta }_M$ in the presence of an external magnetic field applied longitudinally (purple arrow) due to the finite-size effects. The QD levels are broadened due to the coupling $\mathrm{\Gamma }$ with a normal metallic lead (N).

Figure 2

Color scale plots of the DOS of a QD as a function of QD energy level ${\varepsilon }_d$ and spectral frequency $\omega$ for distinct regimes. In all situations we chose $U=5E_0$ and ${\lambda }_L=1.0E_0$. (a) Highly nonlocal isolated MBSs $\left({\delta }_M={\lambda }_R=0\right)$ for $V_Z=0.8E_0$ and $L=2.0\mu \mathrm{m}$; (b) bow-tie profile, wherein $V_Z=1.72E_0,L=0.4\mu \mathrm{m},{\delta }_M=0.12E_0$, and ${\lambda }_R=0.003E_0$. (c) Regular fermionic zero mode $\left({\lambda }_L={\lambda }_R=E_0,{\delta }_M=0\right)$ with $L=0.4\mu \mathrm{m}$ and $V_Z=1.38E_0$. (d)–(f) Diamond profiles $\left({\delta }_M\ll {\lambda }_R,{\lambda }_L\right)$ with actual nanowire length $L=0.4\mu \mathrm{m}$. (d) exhibits the situation for $V_Z=1.4E_0,{\delta }_M=0.004E_0$, and ${\lambda }_R=0.03E_0$. In (e) we have set $V_Z=1.5E_0,{\delta }_M=0.04E_0$, and ${\lambda }_R=0.25E_0$, while in (f) $V_Z=1.6E_0,{\delta }_M=0.08E_0$, and ${\lambda }_R=0.5E_0$.

Figure 3

Color scale plots of the DOS of a QD as a function of QD energy level ${\varepsilon }_d$ and spectral frequency $\omega$ for distinct regimes. In all situations we chose ${\lambda }_L=1.0E_0$. Left panels show (a) isolated MBSs and (c) bow-tie and (e) diamond profiles, corresponding to the cases illustrated by Figs. 2, 2 and 2, respectively, but for the distinct values of Zeeman splitting $V_Z$ and charging energy $U$. Right panels reproduce the corresponding left ones in the reduced gray scale.

Figure 4

Density of states as a function of the QD level ${\varepsilon }_d$ at $\omega =0$ for (a) isolated MBSs and (b) bow-tie and (c) diamond configurations. Plots for several values of charging energy $[U=2.5E_0$ (blue squares), $5E_0$ (magenta triangles) and $7.5E_0$ (green crosses)] are presented.

Figure 5

(a)–(d) Total DOS as a function of $\omega$ for MBSs [corresponding to Fig. 2] for various values of ${\varepsilon }_d$ corresponding to the vertical dashed white lines in Fig. 2. (e)–(h) Spin-resolved DOS ${\rho }_{\sigma }\left(\omega \right)$ for the same parameters as in (a)–(d).

Figure 6

(a)–(d) Total DOS as a function of $\omega$ for the bow-tie profile [corresponding to Fig. 2] for various values of ${\varepsilon }_d$ corresponding to the vertical dashed white lines in Fig. 2. (e)–(h) Spin-resolved DOS ${\rho }_{\sigma }\left(\omega \right)$ for the same parameters as in (a)–(d).

Figure 7

(a)–(d) Total DOS as a function of $\omega$ for the diamond profile for various values of ${\varepsilon }_d$ corresponding to the vertical dashed white lines in Fig. 2. (e)–(h) Spin-resolved DOS ${\rho }_{\sigma }\left(\omega \right)$ for the same parameters as in (a)–(d).

Figure 8

DOS related to AR processes for each spin component, where two representative situations were considered. (a) depicts the same case as in Figs. 6–6, while (b) is related to Figs. 7–7.