Disorder-induced zero-bias peaks in Majorana nanowires

Focusing specifically on the recently retracted work by Zhang et al. [H. Zhang et al., Nature (London) 556, 74 (2018); Retraction, Nature (London) 591, E30 (2021)] and the related recently available correctly analyzed data from this Delft experiment (H. Zhang et al., arXiv:2101.11456), we discuss the general problem of confirmation bias in experiments verifying various theoretical topological quantization predictions. We show that the Delft Majorana experiment is most likely dominated by disorder, which produces trivial (but quite sharp and large) zero-bias Andreev tunneling peaks with large conductance $\sim 2e^2/h$ in the theory, closely mimicking the data. Thus, although the corrected Delft data are by far the best tunnel spectroscopy results available in the literature, manifesting large and sharp zero-bias peaks rising above the background with an impressive hard superconducting gap, our theory shows that the most natural explanation for these zero-bias peaks is that they are disorder induced and not topological Majorana modes. It is possible to misinterpret such disorder-induced zero-bias trivial peaks as the apparent Majorana quantization, as was originally done arising from confirmation bias. One characteristic of the disorder-induced trivial peaks is that they manifest little stability as a function of Zeeman field, chemical potential, and tunnel barrier, distinguishing their trivial behavior from the expected topological robustness of non-Abelian Majorana zero modes. We also analyze a more recent nanowire experiment [P. Yu et al., Nat. Phys. 17, 482 (2021)] which is known to have a huge amount of disorder, showing that such highly disordered nanowires may produce very small above-background trivial peaks with conductance values $\sim 2e^2/h$ in a dirty system manifesting very soft superconducting gap with substantial in-gap conduction, as were already reported by several groups almost 10 years ago. Removing disorder and producing cleaner samples through materials quality improvement and better fabrication is the only way for future progress in this field.

Figure 1

(a) The experimental device from Fig. 1 of Ref. [28]. The InSb nanowire (gray) is covered by the Al shell (green). The tunnel gate (red) controls the tunnel barrier $V_g$ in (b). The super gate (purple) controls the chemical potential. The normal lead (yellow) is attached to one end of the nanowire. (b) The schematic of the semiconductor-superconductor hybrid nanowire in the theory. The bias voltage is applied to the lead and the superconductor is grounded.

Figure 2

The experimental tunneling conductances from Ref. [28]. Panels (a)–(c) are tunneling conductances as a function of the bias voltage and magnetic field at various tunnel gate voltages $V_{\text{TG}}=-7.74$, $-7.80$, $-7.92$ V, respectively. (d) The horizontal line cuts at zero bias as a function of magnetic field corresponding to (a)–(c). (e) The vertical line cuts at zero magnetic field (red) and finite magnetic field (orange) at $V_{\text{TG}}=-7.74$ V. (f) The closed-up view of the conductance peak near $2e^2/h$, indicated by the dashed rectangle in (d). The shading regions indicate the error bar of $1\sigma$ (darker) and $2\sigma$ (lighter). Refer to Ref. [28] for the other parameters of gate voltages.

Figure 3

The theoretical tunneling conductances in the trivial regime which qualitatively reproduce the experimental measurements in Fig. 2. Panels (a)–(c) are tunneling conductances as a function of the bias voltage and Zeeman field at different tunnel barriers $V_g=15.3$, 14.0, and 16.5 meV, respectively. (d) The horizontal line cuts at zero bias as a function of Zeeman field corresponding to (a)–(c). (e) The vertical line cuts at zero magnetic field (red) and finite magnetic field (orange) at $V_g=15.3\mathrm{meV}$. (f) The closed-up view of the conductance peak near $2e^2/h$. Note that the error bar is not applicable here compared to Fig. 2 because it is purely the theoretical calculation. The parameters are as follows: the wire length is $1\mu \mathrm{m}$, the chemical potential $\mu =1\mathrm{meV}$, the variance of disorder ${\sigma }_{\mu }=1\mathrm{meV}$, the parent SC gap ${\mathrm{\Delta }}_0=0.2\mathrm{meV}$ [38], the SC-SM coupling strength $\gamma =0.2\mathrm{meV}$, the parent SC gap closes at $V_z=1.2\mathrm{meV}$, the spin-orbit coupling ${\alpha }_R=0.5$ eV Å, the phenomenological dissipation is $3\times {10}^{-3}\mathrm{meV}$ [39], and zero temperature.

Figure 4

(a) The calculated zero-bias tunnel conductance as a function of the chemical potential $\mu$ at a fixed tunnel barrier $V_g=15.3\mathrm{meV}$ for $V_z=0.8\mathrm{meV}$ (blue), 0.9 meV (orange), 1 meV (red). (b) The calculated zero-bias tunnel conductance as a function of the tunnel barrier $V_g$ at a fixed chemical potential $\mu =1\mathrm{meV}$ for $V_z=$ 0.8 meV (blue), 0.9 meV (orange), 1 meV (red). Refer to Fig. 3 for the rest of the parameters.

Figure 5

The energy spectra (upper panels) and wave functions (lower panels) of the disordered (left panels) and pristine nanowires (right panels) InSb/Al hybrid nanowire. (a) The energy spectrum as a function of Zeeman field in the presence of disorder, which corresponds to Fig. 3. (b) The energy spectrum as a function of Zeeman field in a pristine nanowire for the comparison. The topological regime is where $V_z>1.02\mathrm{meV}$ indicated by the vertical red dashed line. The SC gap always persists as Zeeman field increases. (c) Shows the lowest and second-lowest wave functions in the trivial regime at $V_z=0.9\mathrm{meV}$ corresponding to the black line in (a). (d) Shows the lowest and second-lowest wave functions in the topological regime at $V_z=1.1\mathrm{meV}$ corresponding to the black line in (b). Refer to Fig. 3 for the other parameters (except for the tunnel barrier and dissipation, which are absent here).

Figure 6

The tunneling conductances in the trivial regime as a function of bias voltages at fixed Zeeman fields 0 meV (red), 0.88 meV (blue), and 1.45 meV (black) to reproduce Fig. 3 in Ref. [33] using exactly the same Hamiltonian as in Fig. 3 but only with a different set of SC parameters corresponding NbTiN: the parent SC gap is 3 meV and the SC-SM coupling strength is 1.5 meV (hence the induced SC gap is around 1 meV) [38]. The other parameters are as follows: the wire length is $0.4\mu \mathrm{m}$, the chemical potential is $\mu =5\mathrm{meV}$, the variance of disorder ${\sigma }_{\mu }=20\mathrm{meV}$, parent SC gap closes at $V_z=10\mathrm{meV}$, the spin-orbit coupling ${\alpha }_R=0.5$ eV Å, the phenomenological dissipation is 0.1 meV, and the tunnel barrier height $V_g$ is 10 meV. We also verify that the conductance on the other end [left end in Fig. 8] of the nanowire has no feature, namely, no subgap states emerge and the conductances are almost zero everywhere inside the parent SC gap.

Figure 7

(a) The calculated zero-bias tunnel conductance as a function of the chemical potential $\mu$ at a fixed tunnel barrier $V_g=10\mathrm{meV}$ for $V_z=0\mathrm{meV}$ (red), 0.88 meV (blue), and 1.45 meV (black). (b) The calculated zero-bias tunnel conductance as a function of the tunnel barrier $V_g$ at a fixed chemical potential $\mu =5\mathrm{meV}$ for $V_z=0\mathrm{meV}$ (red), 0.88 meV (blue), 1.45 meV (black). Refer to Fig. 6 for the rest of the parameters.

Figure 8

The energy spectra (upper panels) and wave functions (lower panels) of the disordered (left panels) and pristine nanowires (right panels) in InSb/NbTiN hybrid nanowire. (a) The energy spectrum as a function of Zeeman field in the presence of disorder, which corresponds to Fig. 6. (b) The energy spectrum as a function of Zeeman field in a pristine nanowire for the comparison. The topological regime is where $V_z>5.22\mathrm{meV}$ indicated by the vertical red dashed line. The SC gap always persists as Zeeman field increases. (c) Shows the lowest and second-lowest wave functions in the trivial regime at $V_z=1.45\mathrm{meV}$ corresponding to the black line in (a). (d) Shows the lowest and second-lowest wave functions in the topological regime at $V_z=6.2\mathrm{meV}$ corresponding to the black line in (b). Refer to Fig. 6 for the other parameters (except for the tunnel barrier and dissipation, which are absent here).