Splitting of the zero-bias conductance peak as smoking gun evidence for the existence of the Majorana mode in a superconductor-semiconductor nanowire

Recent observations of a zero-bias conductance peak in tunneling transport measurements in superconductor-semiconductor nanowire devices provide evidence for the predicted zero-energy Majorana modes, but not the conclusive proof of their existence. We establish that direct observation of a splitting of the zero-bias conductance peak can serve as the smoking gun evidence for the existence of the Majorana mode. We show that the splitting has an oscillatory dependence on the Zeeman field (chemical potential) at fixed chemical potential (Zeeman field). By contrast, when the density is constant rather than the chemical potential—the likely situation in the current experimental setups—the splitting oscillations are generically suppressed. Our theory predicts the conditions under which the splitting oscillations can serve as the smoking gun for the experimental confirmation of the elusive Majorana mode.

Figure 1

Oscillatory splitting of the Majorana mode as a function of $V_Z$ (for $\mu ={\mu }_3$, top) and $\Delta \mu =\mu -{\mu }_3$ (for $V_Z=0.75$ meV, bottom). The parameters used in the calculation are $\Delta =0.25$ meV, ${\alpha }^2{m}^{*}/2{\hslash }^2=50\mu \text{eV}$, and $m_{\mathrm{eff}}=0.015m_e$. This behavior is independent of the number of occupied bands. Finite energy resolution ($\delta E=10\mu$eV, yellow band) may prevent the observation of oscillatory slitting in long wires.

Figure 2

Top: Dependence of the SC gap on the Zeeman field in the topological SC phase (yellow) for two different values of the chemical potential. Bottom: Decay length of the MFs as a function of Zeeman splitting. Note that the minimum of the coherence length occurs at $V_Z\gtrsim V_{Z,c}$, right above the critical field, and does not coincide with the maximum of the energy gap.

Figure 3

Comparison between the energy splitting of the Majorana mode at constant chemical potential (orange lines) and constant density (blue lines). The quasiperiodicity of the oscillations at constant $\mu$ is absent in a wire with fixed number of particles. The difference is particularly striking in the single-band case (top panels). Under the constant density condition, small variations of the initial occupancy result in qualitative changes of the dependence of $\Delta E$ on $V_Z$. In contrast, the constant chemical potential condition is characterized by a generic oscillatory splitting.

Figure 4

Left panels: MF splitting as a function of chemical potential $\mu$ and Zeeman field $V_Z$ for single-band system (top) and for a wire with $n_{\mathrm{top}}=3$ (bottom). The dark blue regions correspond to the minima of the energy splitting. The parallel bands represent the source of the oscillatory splitting and a hallmark for Majorana physics. Right panel: Density as a function of chemical potential $\mu$ and Zeeman splitting $V_Z$. Note that, in the single-band case, the constant density contours are nearly parallel with the structures in the left panel, which may result in a strongly suppressed splitting or in a monotonic dependence of the splitting on $V_Z$.