Physical mechanisms for zero-bias conductance peaks in Majorana nanowires

Motivated by the need to understand and simulate the ubiquitous experimentally observed zero-bias conductance peaks in superconductor-semiconductor hybrid structures, we theoretically investigate the tunneling conductance spectra in one-dimensional nanowires in proximity to superconductors in a systematic manner taking into account several different physical mechanisms producing zero-bias conductance peaks. The mechanisms we consider are the presence of quantum dots, inhomogeneous potential, random disorder in the chemical potential, random fluctuations in the superconducting gap, and in the effective $g$ factor with the self-energy renormalization induced by the parent superconductor in both short ($L\sim 1\mu \mathrm{m}$) and long nanowires ($L\sim 3\mu \mathrm{m}$). We classify all foregoing theoretical results for zero-bias conductance peaks into three types: the good, the bad, and the ugly, according to the physical mechanisms producing the zero-bias peaks and their topological properties. We find that, although the topological Majorana zero modes are immune to weak disorder, strong disorder (“ugly”) completely suppresses topological superconductivity and generically leads to trivial zero-bias peaks. Compared qualitatively with the extensive existing experimental results in the superconductor-semiconductor nanowire structures, we conclude that most current experiments are likely exploring trivial zero-bias peaks in the “ugly” situation dominated by strong disorder. We also study the nonlocal end-to-end correlation measurement in both the short and long wires, and point out the limitation of the nonlocal correlation in ascertaining topological properties particularly when applied to short wires. Although we present results for “good” and “bad” zero-bias peaks, arising respectively from topological Majorana bound states and trivial Andreev bound states, strictly for the sake of direct comparison with the “ugly” zero-bias conductance peaks arising from strong disorder, the main goal of the current work is to establish with a very high confidence level the real physical possibility that essentially all experimentally observed zero-bias peaks in Majorana nanowires are most likely ugly, i.e., purely induced by strong disorder, and are as such utterly nontopological. Our work clearly suggests that an essential prerequisite for any future observation of topological Majorana zero modes in nanowires is a substantial materials improvement of the semiconductor-superconductor hybrid systems leading to much cleaner wires.

Figure 1

The schematic of the NS junction composed of a lead and (a) pristine nanowire with a constant SC gap $\mathrm{\Delta }$ in the clean limit $V\left(x\right)=0$; (b) nanowire with a quantum dot $V\left(x\right)$ and a partially covered parent SC; (c) nanowire with an inhomogeneous potential $V\left(x\right)$ and a constant SC gap $\mathrm{\Delta }$; (d) nanowire with disorder $V\left(x\right)$ in the chemical potential; (e) nanowire with disorder $\tilde{g}\left(x\right)$ in the effective $g$ factor; and (f) nanowire with disorder in the SC gap $\mathrm{\Delta }\left(x\right)$.

Figure 2

(a) and (b) show an example of the good ZBCP in a pristine nanowire with the self-energy in a $1\mu \mathrm{m}$ wire. The color plots show the differential tunneling conductance $G$ as a function of $V_{\text{Z}}$ ($x$ axis) and $V_{\text{bias}}$ ($y$ axis) from the left lead (left column) and the right lead (right column). The SC gap collapse $V_{\text{C}}=3$ meV. The TQPT is labeled in the white dashed line at $V_{\text{Z}}=1.02$ meV. The complete correlation conductance measurements are shown in Fig. 11; (c) and (d) show an example of the good ZBCP in the presence of a small amount of disorder in the chemical potential in a $1\mu \mathrm{m}$ wire. The parameters are: standard deviation of disorder in the chemical potential ${\sigma }_{\mu }=0.4$ meV, SC gap collapse $V_{\text{C}}=3$ meV. The TQPT is labeled in the white dashed line at $V_{\text{Z}}=1.02$ meV. The complete correlation conductance measurements are shown in Fig. 12; (e) and (f) show an example of the good ZBCP in the presence of disorder in the SC gap in a $1\mu \mathrm{m}$ wire. The parameters are: standard deviation of disorder in the gap ${\sigma }_{\mathrm{\Delta }}=0.06$ meV, mean parent SC gap ${\mathrm{\Delta }}_0=0.2$ meV, and SC gap co llapse $V_{\text{C}}=3$ meV. The TQPT is labeled in the white dashed line at $V_{\text{Z}}=1.02$ meV. The complete correlation conductance measurements are shown in Fig. 13.

Figure 3

Two examples of the bad ZBCP due to the quantum dot in (a) and (b) and the inhomogeneous potential in (c) and (d) respectively with the self-energy in a $1\mu \mathrm{m}$ wire. The left (right) column shows the conductance measured from the left (right) lead. For the quantum dot case [(a) and (b)], the parameters are: SC gap collapse $V_{\text{C}}=1$ meV, the peak value of the Gaussian-shaped quantum dot $V_{\text{D}}=1.7$ meV, and the size of the quantum dot $l=0.2\mu \mathrm{m}$. For the inhomogeneous potential case [(c) and (d)], the parameters are: SC gap collapse $V_{\text{C}}=1$ meV, the peak value of the Gaussian-shaped potential confinement $V_{\text{max}}=1.4$ meV, and the linewidth $\sigma =0.15\mu \mathrm{m}$. The complete correlation conductance measurements are shown in Fig. 14 for the quantum dot and Fig. 15 for the inhomogeneous potential respectively.

Figure 4

Two examples of the ugly ZBCP in the presence of a large amount of disorder in the chemical potential with the self-energy in the $1\mu \mathrm{m}$ wire. (a) and (b) share a common configuration of disorder; (c) and (d) share another common one. The left(right) column shows the conductance measured from the left(right) lead. The parameters are: standard deviation of disorder in the chemical potential ${\sigma }_{\mu }=1$ meV, SC gap collapse $V_{\text{C}}=1.2$ meV. The nominal TQPT is labeled in the white dashed line at $V_{\text{Z}}=1.02$ meV. The complete correlation conductance measurements are shown in Fig. 16.

Figure 5

Two examples of the ugly ZBCP in the presence of disorder in the effective $g$ factor with the self-energy in the $1\mu \mathrm{m}$ wire. (a) and (b) share a common configuration of disorder; (c) and (d) share another common one. The left (right) column shows the conductance measured from the left (right) lead. The parameters are: standard deviation of disorder in the effective $g$ factor is ${\sigma }_g=0.8$, SC gap collapse $V_{\text{C}}=1.2$ meV. The nominal TQPT is labeled in the white dashed line at $V_{\text{Z}}=1.02$ meV. The complete correlation conductance measurements are shown in Fig. 17.

Figure 6

The conductance spectra as a function of the chemical potential $\mu$ and $V_{\text{bias}}$ at zero magnetic field in the $1\mu \mathrm{m}$ wire with the self-energy. The first (second) row shows the conductance measured from the left (right) lead. Note that the range of the conductance is $0\sim 4e^2/h$ here. (a) and (f) are in the pristine nanowire case. (b) and (g) are in the presence of a quantum dot with the peak value of $V_{\text{D}}=1.7$ meV and the size of $l=0.2\mu \mathrm{m}$. (c) and (h) are in the presence of an inhomogeneous potential with the peak value of $V_{\text{max}}=1.4$ meV and the linewidth $\sigma =0.15\mu \mathrm{m}$. (d), (i), (e), and (j) are in the presence of disorder in the chemical potential with two distinct configurations. The standard deviation of disorder is ${\sigma }_{\mu }=3$ meV.

Figure 7

(a) Tunneling conductance as a function of the magnetic field at a small transmission rate to the lead. The darker color indicates the smaller conductance. This experimental result is from Ref. [28]; (b) fine-tuning parameters to fit (a); The ZBCP is the ugly one with ${\sigma }_{\mu }=1$ meV; (c) tunneling conductance as a function of the magnetic field. The redder color indicates the larger conductance. This experimental result is from Ref. [29]; (d) fine-tuning parameters to fit (d). The ZBCP is the ugly one with ${\sigma }_{\mu }=1$ meV.

Figure 8

Conductance spectra measured from the left lead (the first row) and the right lead (the second row) in the long wire $L=3\mu \mathrm{m}$. (a) and (f) are the good ZBCP in the pristine nanowire with the SC gap collapse $V_{\text{C}}=3$ meV. (b) and (g) are the bad ZBCP in the presence of the quantum dot with the peak value of $V_{\text{D}}=0.6$ meV and the size of $l=0.4\mu \mathrm{m}$. The SC gap collapse is $V_{\text{C}}=1$ meV. (c) and (h) are the bad ZBCP in the presence of the inhomogeneous potential with the peak value of $V_{\text{max}}=1.2$ meV and the linewidth of $\sigma =0.4\mu \mathrm{m}$. The SC gap collapse is $V_{\text{C}}=1$ meV. (d) and (i) are the ugly ZBCP in the presence of disorder in the chemical potential, where ${\sigma }_{\mu }=1$ meV. The SC gap collapse is $V_{\text{C}}=1.2$ meV. (e) and (j) are the ugly ZBCP in the presence of disorder in the effective $g$ factor, where ${\sigma }_g=0.6$. The SC gap collapse is $V_{\text{C}}=1.2$ meV.

Figure 9

(a) The conductance at the zero-bias voltage as a function of the magnetic field. (b) The conductance cut as a function of the bias voltage at the magnetic field of the maximal peak. Both of these experimental results are from Ref. [29].

Figure 10

[(a) and (b)] The conductance at zero-bias voltage and at the Zeeman field of the maximal peak (labeled in black) respectively in an instance of good ZBCP shown in Fig. 11. [(c) and (d)] The conductance at zero-bias voltage and at the Zeeman field of the maximal peak (labeled in black) respectively in an instance of bad ZBCP shown in Fig. 3. [(e) and (f)] The conductance at zero-bias voltage and at the Zeeman field of the maximal peak (labeled in black) respectively in an instance of ugly ZBCP shown in Fig. 4 at the temperature $T=58$ mK. [(g) and (h)] The conductance at zero-bias voltage and at the Zeeman field of the maximal peak (labeled in black) respectively in an instance of ugly ZBCP shown in Fig. 4 at the temperature $T=75$ mK. [(i) and (j)] The conductance at zero-bias voltage and at the Zeeman field of the maximal peak (labeled in black) respectively from the random matrix theory in Ref. [64].

Figure 11

The good ZBCP in two $1\mu \mathrm{m}$ pristine wires [shown in (a)–(d)] and two $3\mu \mathrm{m}$ pristine wires [shown in (c)–(h)]. The color plots show the differential tunneling conductance $G$ as a function of $V_{\text{Z}}$ ($x$ axis) and $V_{\text{bias}}$ ($y$ axis) measured from the left lead (in the first row) and the right lead (in the second row). Nanowires with the self-energy are shown in (b), (f), (d), and (h) and without the self-energy are shown in (a), (c), (e), and (g). The SC gap collapse $V_{\text{C}}=3$ meV for the self-energy case. The TQPT is labeled in the white dashed line at $V_{\text{Z}}=1.02$ meV. The corresponding wave functions and energy spectra are shown in Fig. 19.

Figure 12

The good ZBCP in the presence of a small amount of disorder in the chemical potential for $1\mu \mathrm{m}$ wires [shown in (a), (b), (e), (f), (i), and (j)] and $3\mu \mathrm{m}$ wires [shown in (c), (d), (g), (h), (k), and (l)]. The color plots show the differential tunneling conductance $G$ as a function of $V_{\text{Z}}$ ($x$ axis) and $V_{\text{bias}}$ ($y$ axis) measured from the left lead (in the first row) and the right lead (in the second row). The third row shows the disorder-averaged conductance over 200 samples; the first two rows are the conductance spectra under one specific configuration of disorder. The standard deviation of disorder in the chemical potential ${\sigma }_{\mu }=0.4$ meV for wires both with the self-energy shown in (b), (d), (f), (h), (j), and (l) and without the self-energy shown in (a), (c), (e), (g), (i), and (k). The SC gap collapse $V_{\text{C}}=3$ meV for the self-energy case. The TQPT is labeled in the white dashed line at $V_{\text{Z}}=1.02$ meV. The corresponding wave functions and energy spectra are shown in Fig. 20.

Figure 13

The good ZBCP in the presence of a small amount of disorder in the SC gap for $1\mu \mathrm{m}$ wires [shown in (a), (b), (e), (f), (i), and (j)] and $3\mu \mathrm{m}$ wires [shown in (c), (d), (g), (h), (k), and (l)]. The color plots show the differential tunneling conductance $G$ as a function of $V_{\text{Z}}$ ($x$ axis) and $V_{\text{bias}}$ ($y$ axis) measured from the left lead (in the first row) and the right lead (in the second row). The third row shows the disorder-averaged conductance over 200 samples; the first two rows are the conductance spectra under one specific configuration of disorder. Mean proximity-indcued SC gap $\mathrm{\Delta }=0.2$ meV/parent SC gap ${\mathrm{\Delta }}_0=0.2\mathrm{meV}$ and the standard deviation of disorder in the SC gap ${\sigma }_{\mathrm{\Delta }}=0.06$ meV are for wires both with the self-energy shown in (b), (d), (f), (h), (j), and (l) and without the self-energy shown in (a), (c), (e), (g), (i), and (k). The SC gap collapse $V_{\text{C}}=3$ meV for the self-energy case. The TQPT is labeled in the white dashed line at $V_{\text{Z}}=1.02$ meV. The corresponding wave functions and energy spectra are shown in Fig. 21.

Figure 14

The bad ZBCP due to the quantum dot in two $1\mu \mathrm{m}$ pristine wires [shown in (a), (b), (e), and (f)] and two $3\mu \mathrm{m}$ pristine wires [shown in (c), (d), (g), and (h)]. The color plots show the differential tunneling conductance $G$ as a function of $V_{\text{Z}}$ ($x$ axis) and $V_{\text{bias}}$ ($y$ axis) measured from the left lead (in the first row) and the right lead (in the second row). For the short wire $L=1\mu \mathrm{m}$, the peak value of the quantum dot $V_{\text{D}}=1.7$ meV and size $l=0.2\mu \mathrm{m}$ for wires both with the self-energy shown in (b) and (f) and without the self-energy shown in (a) and (e). For the long wire $L=3\mu \mathrm{m}$, the peak value of the quantum dot $V_{\text{D}}=0.6$ meV and size $l=0.4\mu \mathrm{m}$ for wires both with the self-energy shown in (d) and (h) and without the self-energy shown in (c) and (g). The SC gap collapse $V_{\text{C}}=1$ meV for the self-energy case. The TQPT is labeled in the white dashed line at $V_{\text{Z}}=1.02$ meV. The corresponding wave functions and energy spectra are shown in Figs. 22 and 23.

Figure 15

The bad ZBCP due to the inhomogeneous potential in two $1\mu \mathrm{m}$ pristine wires [shown in (a), (b), (e), and (f)] and two $3\mu \mathrm{m}$ pristine wires [shown in (c), (d), (g), and (h)]. The color plots show the differential tunneling conductance $G$ as a function of $V_{\text{Z}}$ ($x$ axis) and $V_{\text{bias}}$ ($y$ axis) measured from the left lead (in the first row) and the right lead (in the second row). For the short wire $L=1\mu \mathrm{m}$, the peak value of the inhomogeneous potential $V_{\text{max}}=1.4$ meV and linewidth $\sigma =0.15\mu \mathrm{m}$ for wires both with the self-energy shown in (b) and (f) and without the self-energy shown in (a) and (e). For the long wire $L=3\mu \mathrm{m}$, the peak value of the inhomogeneous potential $V_{\text{max}}=1.2$ meV and linewidth $\sigma =0.4\mu \mathrm{m}$ for wires both with the self-energy shown in (d) and (h) and without the self-energy shown in (c) and (g). The SC gap collapse $V_{\text{C}}=1$ meV for the self-energy case. The TQPT is labeled in the white dashed line at $V_{\text{Z}}=1.02$ meV. The corresponding wave functions and energy spectra are shown in Figs. 24 and 25.

Figure 16

The ugly ZBCP in the presence of a large amount of disorder in the chemical potential for two $1\mu \mathrm{m}$ wires (shown in the first two rows) and for one $3\mu \mathrm{m}$ wire (shown in the third row). The color plots show the differential tunneling conductance $G$ as a function of $V_{\text{Z}}$ ($x$ axis) and $V_{\text{bias}}$ ($y$ axis) measured from the left lead (shown in the first and third columns) and the right lead (shown in the second and fourth columns). The conductance spectra in the first row share a common configuration of disorder; the ones in the second row share another. The standard deviation of the chemical potential ${\sigma }_{\mu }=1$ meV for wires both with the self-energy shown in (c), (d), (g), and (h), and without the self-energy shown in (a), (b), (e), and (f). For $L=3\mu \mathrm{m}$ wire, the standard deviation of the chemical potential ${\sigma }_{\mu }=1$ for wires both with the self-energy shown in (k) and (l) and without the self-energy shown in (i) and (j). The SC gap collapse $V_{\text{C}}=1.2$ meV for the self-energy case. The TQPT is labeled in the white dashed line at $V_{\text{Z}}=1.02$ meV. The corresponding wave functions and energy spectra are shown in Figs. 26 and 27.

Figure 17

The ugly ZBCP in the presence of disorder in the effective $g$ factor for two $1\mu \mathrm{m}$ wires (shown in the first two rows) and for one $3\mu \mathrm{m}$ wire (shown in the third row). The color plots show the differential tunneling conductance $G$ as a function of $V_{\text{Z}}$ ($x$ axis) and $V_{\text{bias}}$ ($y$ axis) measured from the left lead (shown in the first and third columns) and the right lead (shown in the second and fourth columns). The conductance spectra in the first row share a common configuration of disorder; the ones in the second row share another. The standard deviation of disorder in the effective $g$ factor is ${\sigma }_g=0.8$ for wires both with self-energy shown in (c), (d), (g), and (h) and without the self-energy shown in (a), (b), (e), and (f). For $L=3\mu \mathrm{m}$ wire, the effective $g$ factor is ${\sigma }_g=0.6$ for wires both with the self-energy shown in (k) and (l) and without the self-energy shown in (i) and (j). The SC gap collapse $V_{\text{C}}=1.2$ meV for the self-energy case. The TQPT is labeled in the white dashed line at $V_{\text{Z}}=1.02$ meV. The corresponding wave functions and energy spectra are shown in Figs. 28 and 29.

Figure 18

The short but uncorrelated occasions in bad (the first and second rows) and ugly (the third row) ZBCPs. The color plots show the differential tunneling conductance $G$ as a function of $V_{\text{Z}}$ ($x$ axis) and $V_{\text{bias}}$ ($y$ axis) measured from the left lead (in the first and third columns) and the right lead (in the second and fourth columns). The bad ZBCPs due to the two quantum dots at both ends in $1\mu \mathrm{m}$ wires are shown in (a)–(d). The peak value of the quantum dot is $V_{\text{D,L}}=1.7$ meV on the left and $V_{\text{D,R}}=2.3$ meV on the right, the size of the quantum dot is $l_{\text{L}}=0.2\mu \mathrm{m}$ on the left and $l_{\text{R}}=0.15\mu \mathrm{m}$ on the right for both wires with the self-energy shown in (c) and (d) and without the self-energy shown in (a) and (b). The SC collapse $V_{\text{C}}=1$ meV. The bad ZBCPs due to the inhomogeneous potential with the Gaussian barriers on both ends are shown in (e), (f), (g), and (h). The peak value of the Gaussian barrier is $V_{\text{max,L}}=1.4$ meV on the left and $V_{\text{max,R}}=1.9$ meV on the right, the linewidth of the Gaussian barrier is ${\sigma }_{\text{L}}=0.15\mu \mathrm{m}$ on the left and ${\sigma }_{\text{R}}=0.1\mu \mathrm{m}$ on the right for both wires with the self-energy shown in (g) and (h) and without the self-energy shown in (e) and (f). The SC collapse $V_{\text{C}}=1$ meV. The ugly ZBCPs due to the disorder in the chemical potential are shown in (i), (j), (k), and (l). The standard deviation of the chemical potential ${\sigma }_{\mu }=1$ meV for wires both with the self-energy shown in (k) and (l) and without the self-energy shown in (i) and (j). The SC collapse $V_{\text{C}}=1.2$ meV. The corresponding wave functions and energy spectra are shown in Fig. 30.

Figure 19

(a)–(d) correspond to Figs. 11 and 11. (e)–(h) correspond to Figs. 11 and 11. (i)–(l) correspond to Figs. 11 and 11. (m)–(p) correspond to Figs. 11 and 11.

Figure 20

(a)–(d) correspond to Figs. 12 and 12. (e)–(h) correspond to Figs. 12 and 12. (i)–(l) correspond to Figs. 12 and 12. (m)–(p) correspond to Figs. 12 and 12.

Figure 21

(a)–(d) correspond to Figs. 13 and 13. (e)–(h) correspond to Figs. 13 and 13. (i)–(l) correspond to Figs. 13 and 13. (m)–(p) correspond to Figs. 13 and 13.

Figure 22

(a)–(d) correspond to Figs. 14 and 14. (e)–(h) correspond to Figs. 14 and 14.

Figure 23

(a)–(d) correspond to Figs. 14 and 14. (e)–(h) correspond to Figs. 14 and 14.

Figure 24

(a)–(d) correspond to Figs. 15 and 15. (e)–(h) correspond to Figs. 15 and 15.

Figure 25

(a)–(d) correspond to Figs. 15 and 15. (e)–(h) correspond to Figs. 15 and 15.

Figure 26

(a)–(d) correspond to Figs. 16 and 16. (e)–(h) correspond to Figs. 16 and 16. (i)–(l) correspond to Figs. 16 and 16. (m)–(p) correspond to Figs. 16 and 16.

Figure 27

(a)–(d) correspond to Figs. 16 and 16. (e)–(h) correspond to Figs. 16 and 16.

Figure 28

(a)–(d) correspond to Figs. 17 and 17. (e)–(h) correspond to Figs. 17 and 17. (i)–(l) correspond to Figs. 17 and 17. (m)–(p) correspond to Figs. 17 and 17.

Figure 29

(a)–(d) correspond to Figs. 17 and 17. (e)–(h) correspond to Figs. 17 and 17.

Figure 30

(a)–(d) correspond to Figs. 18 and 18. (e)–(h) correspond to Figs. 18 and 18. (i)–(l) correspond to Figs. 18 and 18. (m)–(p) correspond to Figs. 18 and 18. (q)–(t) correspond to Figs. 18 and 18. (u)–(x) correspond to Figs. 18 and 18.