Detecting a Majorana-fermion zero mode using a quantum dot

We propose an experimental setup for detecting a Majorana zero mode consisting of a spinless quantum dot coupled to the end of a $p$-wave superconducting nanowire. The Majorana bound state at the end of the wire strongly influences the conductance through the quantum dot: Driving the wire through the topological phase transition causes a sharp jump in the conductance by a factor of $1/2$. In the topological phase, the zero-temperature peak value of the dot conductance (i.e., when the dot is on resonance and symmetrically coupled to the leads) is $e^2/2h$. In contrast, if the wire is in its trivial phase, the conductance peak value is $e^2/h$, or if a regular fermionic zero mode occurs on the end of the wire, the conductance is 0. The system can also be used to tune Flensberg's qubit system [Phys. Rev. Lett. 106, 090503 (2011)] to the required degeneracy point.

Figure 1

(a) Sketch of a dot-MBS system: the semiconductor wire on an $s$-wave superconductor surface, and a magnetic field perpendicular to the surface ($\widehat{z}$ direction). The dot couples to one end of the wire; the conductance through the dot is measured by adding two external leads. (b) Majorana chain representation for a leads-dot-MBS system ($G_{\mathrm{peak}}=e^2/2h$). (c) Dot-leads system with nothing side coupled (left) and a Majorana chain representation (right) ($G_{\mathrm{peak}}=e^2/h$). (d) Dot-leads system with side-coupled regular fermionic zero mode (left) and Majorana chain representation (right) ($G_{\mathrm{peak}}=0$).

Figure 2

Spectral function of the quantum dot in the on-resonance (${\epsilon }_d=0$) and symmetric (${\Gamma }_L={\Gamma }_R=\Gamma /2$) case. (a) Coupling from the dot to MBS ($\lambda$) and leads ($\Gamma$) varies at fixed ${\epsilon }_M=0$. Solid lines: $\Gamma =0.2$ and $\lambda$ from 0 to $0.1$. Dashed lines: $\lambda =0.02$ and $\Gamma$ from $0.05$ to $0.1$. The spectral function evolves from a simple resonant tunneling form in the absence of coupling to a three-peak structure; the middle peak is a direct result of the Majorana zero mode. (b) MBS-MBS coupling strength varies at fixed $\Gamma =0.2$, $\lambda =0.1$. Note that $A(\omega =0)=1/2$ whenever a Majorana is coupled. The unit is chosen so that the lead band width is $D_L=40$ for all calculations.

Figure 3

Dot spectral function and peak conductance in the more realistic nanowire case (the dot is on resonance and symmetrically coupled to the probe leads). $A\left(\omega \right)$ for different values of (a) the SC order parameter at fixed ${\alpha }_R=2$ and (b) the Rashba interaction strength at fixed $\Delta =3$. The results are qualitatively similar to those of the simple model (Fig. 2). (Parameters: $\Gamma =0.1$, $\lambda =0.3$, and $V_z=6$.) (c),(d) Conductance as a function of Zeeman energy for different temperatures at fixed $\Delta =3$. The sharp change at $V_z=\Delta$ is a signature of the topological phase transition. [Parameters: (c) ${\alpha }_R=2$, $\lambda =0.1$, $\Gamma =0.1$; (d) ${\alpha }_R=10$, $\lambda =0.3$, $\Gamma =0.08$.] Throughout, $\mu =0$, the hopping in the nanowire $t=10$ corresponds to a bandwidth $D=40$, and the wire consists of 1000 sites.

Figure 4

(a) Conductance for a MBS-dot-MBS system as a function of the phase $\phi =\Phi /{\Phi }_0$ for different temperatures; the dot is on resonance and symmetrically coupled to the STM tips. ($T=0.01$, $0.005$, $0.00125$, and 0, from top to bottom; parameters are ${\lambda }_1={\lambda }_2={\Gamma }_1={\Gamma }_2=0.1$.) This curve does not depend on the value of $|{\lambda }_1/{\lambda }_2|$. (b) Sketch of a MBS-dot-MBS system. The two MBSs appear at the ends of the nanowire; $\Phi$ is the magnetic flux through the loop. The conductance is measured using a dual-tip STM, allowing tuning to the degeneracy point.