A study on the tunneling spectroscopy of an ${\rm N}-p{\rm S}$ junction and an ${\rm N}-{\rm hS}$ junction

We first consider the one-dimensional ${\rm N}-p{\rm S}$ junction shown in figure 1(a). Under the representation ${{\Psi }^{\dagger }}(x)=({{\psi }^{\dagger }}(x),\psi (x))$, the Hamiltonian is given as

$$\begin{eqnarray}&&H=\left[ -\frac{{{\hbar }^{2}}}{2\;m}\frac{{{{\rm d}}^{2}}}{{\rm d}{{x}^{2}}}-\mu (x)+V(x)+H\delta (x) \right]{{\sigma }_{z}}+\Delta (x){{\sigma }_{x}},\end{eqnarray} \tag{ 1 }$$

where $\vec{\sigma }=({{\sigma }_{x}},{{\sigma }_{y}},{{\sigma }_{z}})$ are Pauli matrices, $H\delta (x)$ is the scattering potential at the interface, V(x) is potential induced by disorder, external field, etc, for the ${\rm N}-p{\rm S}$ junction, and we set $V(x)=0$. $\mu (x)$ is the chemical potential, and we set $\mu (x)={{\mu }_{\,n}}$ for $x\lt 0$ and $\mu (x)={{\mu }_{\,s}}$ for $0\lt x\lt L$, $\delta \mu ={{\mu }_{\,n}}-{{\mu }_{\,s}}$ is the chemical mismatch (note μ_s only has the meaning of chemical potential, when superconductivity is absent). $\Delta (x)$ is the paring potential, which should be determined self-consistently [54]; however, here we assume $\Delta (x)=-{\rm i}{{\Delta }_{p}}\theta (x)\theta (L-x)\hbar {{\partial }_{x}}$ for simplicity, which means that Δ_p is a constant in the region $0\lt x\lt L$ . For simplicity, in this work we also neglect the effects of disorder, such as impurities that will induce pair-breaking and bound states in the gap [55].

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For $0\lt x\lt L$, the p-wave superconductor, the Hamiltonian in momentum space is given as (in the following, we set $\hbar =m=1$)

$$\begin{eqnarray}&&{{H}_{S}}=\left[ \frac{{{k}^{2}}}{2}-{{\mu }_{\,s}} \right]{{\sigma }_{z}}+{{\Delta }_{p}}k{{\sigma }_{x}},\end{eqnarray} \tag{ 2 }$$

which is the continuum form of the Kitaev model [6]. Its energy spectrum ${{E}_{k}}=\pm \sqrt{{{(\frac{{{k}^{2}}}{2}-{{\mu }_{\,s}})}^{2}}+{{({{\Delta }_{p}}k)}^{2}}}$. It is easy to see that the energy gap is only closed at ${{\mu }_{\,s}}=0$. The critical point ${{\mu }_{\,c}}=0$ separates two physical regimes: the weak-pairing (${{\mu }_{\,s}}\gt 0$) and strong-pairing (${{\mu }_{\,s}}\lt 0$) phases. The two distinct names come from the fact that for ${{\mu }_{\,s}}\gt 0$, without pairing, the system is metallic, and therefore the resulting superconducting phase is BCS-like, while for ${{\mu }_{\,s}}\lt 0$, without pairing, the system is an insulator and therefore the resulting superconducting phase obviously does not fit the BCS picture.

One of the two phases is a topological superconducting phase with zero-energy Majorana fermions localized at the end of the system. The topology of a system is usually determined by the topology of its momentum space [56]. To determine which phase is a topological superconducting phase, here we show the energy spectrums of these two superconducting phases with open boundaries. Figure 2(a) shows that when the lattice model parameters satisfy ${{\mu }_{\,s}}\gt -2t$ (equivalent to ${{\mu }_{\,s}}\gt 0$ in the continuum model), there is a pair of zero-energy states (two red points) in the gap, which suggests this phase (the weak-pairing phase) is a topological superconducting phase. When the system is sufficiently long, the two zero modes are well separated and localized at the two ends of the system and, as a result, the wave functions of the two modes do not overlap. Figure 2(b) shows that when ${{\mu }_{\,s}}\lt -2t$ (equivalent to ${{\mu }_{\,s}}\lt 0$ in the continuum model), no zero energy modes appear in the gap and, as a result, the density profile exhibited is not localized at the end of the system. This phase is the normal superconducting phase. However, when the system is not sufficiently long, the wave functions of the two modes will overlap and, consequently, the energy of the end states will split and no longer be zero. The shorter the system is, the larger the overlap and the split are, as shown in figures 2(c) and (d).

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As shown above, when μ_s goes across the critical point ${{\mu }_{\,c}}=0$, the system undergoes a topological phase transition. To study the behavior of tunneling spectroscopy when μ_s goes across the critical point ${{\mu }_{\,c}}=0$, we consider $\Delta _{p}^{2}\gt 2{{\mu }_{\,s}}$, which will restrict μ_s to be very close to μ_c and the minimum of the energy spectrum is located at k = 0. For a given energy E (here we only consider in-gap states, i.e. $E\lt {{E}_{g}}/2=|{{\mu }_{\,s}}|$, as the differential-tunneling conductance at zero-bias voltage), the wave function in the p-wave superconductor is given as

$$\begin{eqnarray}\begin{array}{rcl} {{\psi }_{S}}(x) & = & c(E)\left( \begin{array}{ccccccccccccccc} {\rm i}{{u}_{+}} \\ {{v}_{+}} \\ \end{array} \right){{{\rm e}}^{-{{k}_{+}}x}}+\tilde{c}(E)\left( \begin{array}{ccccccccccccccc} -{\rm i}{{u}_{+}} \\ {{v}_{+}} \\ \end{array} \right){{{\rm e}}^{{{k}_{+}}x}} \\ {} & {} & +d(E)\left( \begin{array}{ccccccccccccccc} {\rm i}{{u}_{-}} \\ {{v}_{-}} \\ \end{array} \right){{{\rm e}}^{-{{k}_{-}}x}}+\tilde{d}(E)\left( \begin{array}{ccccccccccccccc} -{\rm i}{{u}_{-}} \\ {{v}_{-}} \\ \end{array} \right){{{\rm e}}^{{{k}_{-}}x}}, \\ \end{array}\end{eqnarray} \tag{ 3 }$$

where ${{k}_{\pm }}=\sqrt{2({{\Delta }^{2}}-{{\mu }_{\,s}})\pm 2\sqrt{{{E}^{2}}+\Delta _{p}^{2}(\Delta _{p}^{2}-2{{\mu }_{\,s}})}}$, ${{u}_{\pm }}={{\Delta }_{p}}{{k}_{\pm }}$, and ${{v}_{\pm }}=E+\frac{k_{\pm }^{2}}{2}+\mu$. c(E), $\tilde{c}(E)$, d(E) and $\tilde{d}(E)$ are energy-dependent coefficients, and when $L=\infty$, $\tilde{c}(E)=\tilde{d}(E)=0$, it is obvious that the wave function is localized at the left end of the wire.

For $x\lt 0$, the normal metal lead, the Hamiltonian is

$$\begin{eqnarray}&&{{H}_{N}}=\left[ \frac{{{k}^{2}}}{2}-{{\mu }_{\,n}} \right]{{\sigma }_{z}};\end{eqnarray} \tag{ 4 }$$

here we keep σ_z for convenience. We consider that an electron is injected from the normal lead into the p-wave superconductor, and the wave function in the normal lead is given as

$$\begin{eqnarray}&&{{\psi }_{N}}(x)=\left( \begin{array}{ccccccccccccccc} {{{\rm e}}^{2{\rm i}{{q}_{e}}x}}+b(E) \\ 0 \\ \end{array} \right){{{\rm e}}^{-{\rm i}{{q}_{e}}x}}+a(E)\left( \begin{array}{ccccccccccccccc} 0 \\ 1 \\ \end{array} \right){{{\rm e}}^{{\rm i}{{q}_{h}}x}},\end{eqnarray} \tag{ 5 }$$

where ${{q}_{e,h}}=\sqrt{2({{\mu }_{\,n}}\pm E)}$. a(E) and b(E) denote the equal-spin Andreev reflection amplitude and normal reflection amplitude, respectively. (Here we assume the Cooper pairs are composed of electrons with the same spin polarization. With this assumption, the spin-reversed Andreev reflection and spin-reversed normal reflection are naturally absent.)

Following the BTK methods [53], the two wave functions (3) and (5) need to satisfy the boundary conditions,

$$\begin{eqnarray}\begin{array}{lll} {} & {} & {{\psi }_{S}}(x=L)=0; \\ {} & {} & {{\psi }_{S}}(x=0)={{\psi }_{N}}(x=0); \\ {} & {} & {{v}_{s}}{{\psi }_{S}}(x={{0}^{+}})-{{v}_{n}}{{\psi }_{N}}(x={{0}^{-}})=-{\rm i}Z{{\sigma }_{z}}{{\psi }_{N}}(x=0), \\ \end{array}\end{eqnarray} \tag{ 6 }$$

where $Z=2\;H$, ${{v}_{s}}=\partial {{H}_{S}}/\partial k$ and ${{v}_{n}}=\partial {{H}_{N}}/\partial k$, two 2 × 2 matrices, are the velocity operators [57]. From equation (6), we can obtain a(E) and b(E) directly, and according to [53], the zero-temperature differential-tunneling conductance is given as

$$\begin{eqnarray}&&G(eV)=\frac{{{e}^{2}}}{h}\left[ 1+A(eV)-B(eV) \right],\end{eqnarray} \tag{ 7 }$$

where $A(E)=|a(E){{|}^{2}}{{q}_{h}}/{{q}_{e}}$, (E = eV), is the equal-spin Andreev reflection probability, and $B(E)=|b(E){{|}^{2}}$ is the normal reflection probability. Here, as we only consider in-gap states, the waves expressed in equation (3) do not carry current, and A(eV) and B(eV) satisfy the normalization condition, i.e. $A(eV)+B(eV)=1$. For different parameters μ_s , L and Z (we set ${{\mu }_{\,n}}=1$ as the unit), the results are shown in figure 3.

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Figures 3(a) and (b) show the important difference of differential-tunneling conductance between ${{\mu }_{\,s}}\gt 0$ (topologically non-trivial) and ${{\mu }_{\,s}}\lt 0$ (topologically trivial). For ${{\mu }_{\,s}}\gt 0$, a quantized zero-bias conductance peak of $2{{e}^{2}}/h$ stably exists (as shown in figures 3(a), (b) and (d)). A stable and quantized zero-bias conductance peak is a manifestation of perfect equal-spin Andreev reflection and indicates a stable topological phase. This result is the same as the result obtained by an 'interface electron-Majorana coupling' model [22, 23]; therefore, even though we do not assume the existence of the Majorana-bound states at the wire end, the non-trivial topology of the p-wave superconductor manifests itself in the tunneling spectroscopy obtained by matching the wave functions of the two bulks directly. For ${{\mu }_{\,s}}\lt 0$, the in-gap differential-tunneling conductance almost vanishes. The important difference indicates that, for the p-wave superconductor, the tunneling experiments can detect the topological quantum transition at ${{\mu }_{\,s}}=0$ effectively. Figures 3(c) and (d) show that increasing the interface scattering potential only narrows the width of the peak; it neither changes the quantized height (an analytical proof for this is given in the appendix) nor the position of the peak. The wire length has a strong impact on the conductance for ${{\mu }_{\,s}}\gt 0$, as shown in figures 3(e) and (f), for L = 40 and L = 60, the conductance peaks appear at finite-bias voltage and the peak will move left (zero-bias voltage) with increasing L (this agrees with the usual two end-Majorana coupled picture, and also agrees with the results shown in figures 2(c) and (d)). The effect of the interface-scattering potential for finite length is similar to that with infinity length mentioned above, as shown in figures 3(g) and (h). For sufficiently short L, for example, $L\lt 10$, there is no conductance peak and the in-gap differential-tunneling conductance almost vanishes. For ${{\mu }_{\,s}}\lt 0$, the topologically trivial phase, the wire length L and the interface scattering potential Z has little effect, the in-gap differential-tunneling conductance remains very small.

Above we have restricted μ_s to be close to μ_c , and there exists a significant mismatch between μ_n and μ_s . Canceling this restriction, we find, for ${{\mu }_{\,s}}\gt 0$, increasing μ_s greatly widens the width of the zero-bias peak of the in-gap tunneling spectroscopy; however, the peakʼs quantization behavior does not change and, therefore, the main physics does not change, as shown in figure 4(a). For ${{\mu }_{\,s}}\lt 0$, compared figures 4(b) to 3(b), it is easy to see that decreasing μ_s has little effect on the in-gap tunneling spectroscopy.

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The one-dimensional ${\rm N}-{\rm hS}$ junction is shown in figure 1(b). For $x\lt 0$, the Hamiltonian is a generalized 4 × 4 matrix form of equation (4). For $0\lt x\lt L$, the Hamiltonian is now given as [11] (in momentum space, under representation $\Psi _{k}^{\dagger }=(c_{k,\uparrow }^{\dagger },c_{k,\downarrow }^{\dagger },{{c}_{-k,\downarrow }},-{{c}_{-k,\uparrow }})$)

$$\begin{eqnarray}&&{{H}_{hS}}={{\xi }_{k}}{{\tau }_{z}}+B{{\sigma }_{x}}+\alpha k{{\sigma }_{y}}{{\tau }_{z}}+{{\Delta }_{s}}{{\tau }_{x}},\end{eqnarray} \tag{ 8 }$$

where ${{\xi }_{k}}={{k}^{2}}/2-{{\tilde{\mu }}_{\,s}}$, B is the in-plane magnetic field along the wire, α is the spin–orbit coupling strength and Δ_s is the s-wave pair potential. $\vec{\tau }=({{\tau }_{x}},{{\tau }_{y}},{{\tau }_{z}})$ and $\vec{\sigma }=({{\sigma }_{x}},{{\sigma }_{y}},{{\sigma }_{z}})$ are Pauli matrices in particle-hole space and spin space, respectively. The heterostructure described by this Hamiltonian is the one that is realized in the experiment [39]. The quasiparticle energy spectrum is given as

$$\begin{eqnarray*}&&E_{k}^{\pm }=\sqrt{\xi _{k}^{2}+{{(\alpha k)}^{2}}+\Delta _{s}^{2}+{{B}^{2}}\pm 2\sqrt{\xi _{k}^{2}{{\alpha }^{2}}{{k}^{2}}+{{B}^{2}}(\xi _{k}^{2}+\Delta _{s}^{2})}},\end{eqnarray*}$$

where the energy gap ${{\Delta }_{g}}\equiv {{\{E_{k}^{-}\}}_{{\rm min} }}$ is closed (${{\Delta }_{g}}=0$) when the magnetic field B reaches the critical value ${{B}_{c}}=\sqrt{\tilde{\mu }_{s}^{2}+\Delta _{s}^{2}}$, which separates $B\lt \sqrt{\tilde{\mu }_{s}^{2}+\Delta _{s}^{2}}$, the topologically trivial phase, from $B\gt \sqrt{\tilde{\mu }_{s}^{2}+\Delta _{s}^{2}}$, the topologically non-trivial phase. The energy–momentum relation here is much more complicated than that of the p-wave superconductor. This complication makes it impossible to write down the analytical form of the wave function ${{\psi }_{hS}}(x)$ directly, so we must seek help from numerical tools.

As the Hamiltonian (8) is a 4 × 4 matrix, the wave function in the normal metal is a four-component vector, which takes the form

$$\begin{eqnarray}&&{{\psi }_{N}}(x)=\left( \begin{array}{ccccccccccccccc} 1 \\ 0 \\ 0 \\ 0 \\ \end{array} \right){{{\rm e}}^{{\rm i}{{q}_{e}}x}}+\left( \begin{array}{ccccccccccccccc} {{b}_{\uparrow }} \\ {{b}_{\downarrow }} \\ 0 \\ 0 \\ \end{array} \right){{{\rm e}}^{-{\rm i}{{q}_{e}}x}}+\left( \begin{array}{ccccccccccccccc} 0 \\ 0 \\ {{a}_{\downarrow }} \\ {{a}_{\uparrow }} \\ \end{array} \right){{{\rm e}}^{{\rm i}{{q}_{h}}x}},\end{eqnarray} \tag{ 9 }$$

where ${{b}_{\uparrow }}(E)$ denotes the normal reflection amplitude, ${{b}_{\downarrow }}(E)$ denotes the spin-reversed reflection amplitude, ${{a}_{\uparrow }}(E)$ denotes the equal-spin Andreev reflection amplitude, and ${{a}_{\downarrow }}(E)$ denotes the spin-reversed Andreev reflection amplitude. Now, the boundary conditions take the form

$$\begin{eqnarray}\begin{array}{lll} {} & {} & {{\psi }_{hS}}(x=L)=0; \\ {} & {} & {{\psi }_{hS}}(x=0)={{\psi }_{N}}(x=0); \\ {} & {} & {{v}_{hs}}{{\psi }_{hS}}(x={{0}^{+}})-{{v}_{n}}{{\psi }_{N}}(x={{0}^{-}})=-{\rm i}Z{{\sigma }_{0}}{{\tau }_{z}}{{\psi }_{N}}(x=0), \\ \end{array}\end{eqnarray} \tag{ 10 }$$

where ${{v}_{hs}}=\partial {{H}_{hS}}/\partial k$ is a 4 × 4 matrix, and ${{v}_{n}}=-{\rm i}{{\partial }_{x}}{{\sigma }_{0}}{{\tau }_{Z}}$ is also generalized to 4 × 4 matrix.

Based on equation (10), we can obtain ${{b}_{\uparrow ,\downarrow }}(E)$ and ${{a}_{\uparrow ,\downarrow }}(E)$, and the differential-tunneling conductance is given as

$$\begin{eqnarray}&&G(eV)=\frac{{{e}^{2}}}{h}\left[ 1+{{A}_{\uparrow }}(eV)+{{A}_{\downarrow }}(eV)-{{B}_{\uparrow }}(eV)-{{B}_{\downarrow }}(eV) \right],\end{eqnarray} \tag{ 11 }$$

where ${{A}_{\uparrow ,\downarrow }}(eV)=|{{a}_{\uparrow ,\downarrow }}(eV){{|}^{2}}{{q}_{h}}/{{q}_{e}}$, and ${{B}_{\uparrow ,\downarrow }}(eV)=|{{b}_{\uparrow ,\downarrow }}(eV){{|}^{2}}$. ${{A}_{\uparrow ,\downarrow }}(eV)$ and ${{B}_{\uparrow ,\downarrow }}(eV)$ in the gap region should satisfy the normalization condition:

$$\begin{eqnarray*}&&{{A}_{\uparrow }}(eV)+{{A}_{\downarrow }}(eV)+{{B}_{\uparrow }}(eV)+{{B}_{\downarrow }}(eV)=1.\end{eqnarray*}$$

For different parameters, the results are shown in figure 5.

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Figures 5(a) and (b) show that when $B\gt \sqrt{\tilde{\mu }_{s}^{2}+\Delta _{s}^{2}}$, in the topological region, a zero-bias conductance peak is formed. However, contrary to the quantized zero-bias conductance peak of an ${\rm N}-p{\rm S}$ junction, here the zero-bias peak is non-quantized and sensitive to parameters. Increasing the interface-scattering potential not only narrows the width of the peak, but also generally increases the height of the peak, as shown in figures 5(c) and (d). From figures 5(a) and (b), we can also see that the increase of the peak height is due to a suppression of the spin-reversed Andreev reflection and a simultaneous increase of the equal-spin Andreev reflection by the interface-scattering potential. However, with a further increase of the interface-scattering potential, this corresponding increasing effect will finally become saturated, and the zero-bias conductance peak will still have a gap to the quantized value, as shown in figure 5(d). For comparison, the inset in figure 5(d) shows that when $B\lt \sqrt{\tilde{\mu }_{s}^{2}+\Delta _{s}^{2}}$, the normal phase region, the zero-bias conductance monotonously decreases with increasing scattering potential.

For comparison, figures 5(e) and (f) shows that when $B\lt \sqrt{\tilde{\mu }_{s}^{2}+\Delta _{s}^{2}}$, the normal phase, no zero-bias peak appears, and by increasing the interface-scattering potential, the normal reflection is greatly enhanced and the spin-reversed Andreev reflection and the conductance are greatly reduced, which is a familiar phenomenon in ${\rm N}-{\rm S}$ junctions [53]. Comparing figures 5(a) and 5(b) to (c) and (d), it is not hard to find that when the system goes from the normal phase to the topological phase, the probability of an equal-spin Andreev reflection is greatly enhanced (but still has a considerable gap to the perfect equal-spin Andreev reflection) and the spin-reversed Andreev reflection amplitude is greatly reduced, which indicates the equal-spin pairing (p-wave pairing) becomes much more favored in the topological region than in the normal region.

Spin–orbit coupling also has a strong impact on tunneling spectroscopy. As shown in figure 6(a), when decreasing the spin–orbit coupling and fixing other parameters, the height of the zero-bias peak and the probability of an equal-spin Andreev reflection occurring are significantly reduced. A further inspection shows that once the spin–orbit coupling decreases to a critical value (named α_g ) not only does the peak height keep decreasing to a smaller value but the energy gap ${{\Delta }_{g}}$ also begins to depend on spin–orbit coupling (when $\alpha \lt {{\alpha }_{g}}$, we say the spin–orbit coupling is weak), with a dependence that monotonically decreases as the spin–orbit coupling decreases. When the spin–orbit coupling is decreased to zero and the gap closes, the system becomes gapless and the zero-bias conductance peak disappears, which indicates a breakdown of the topological criterion.

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Although we have shown that decreasing the spin–orbit coupling (from the figure 5 parameter, $\alpha =0.5$) lowers the peak, we find that increasing the spin–orbit coupling does not correspond to a monotonic increase of the peak height. As shown in figures 6(b) and (c), the zero-bias conductance peak first increases and then decreases with increasing spin–orbit coupling, the optimal spin–orbit coupling α_c under the parameters given in figure 6(c) is about 0.6. In the following, when ${{\alpha }_{g}}\lt \alpha \lt {{\alpha }_{c}}$, we say that the spin–orbit coupling is in the intermediate region, and when $\alpha \gt {{\alpha }_{c}}$, we say that the spin–orbit coupling is strong. From figure 6(c), we also find when α goes beyond α_c , a larger α corresponds to a lower peak and the reduction effect due to the increase of spin–orbit coupling is significant. However, we also find, almost simultaneously, that when α goes beyond α_c , the saturation effect of the interface scattering potential for weak spin–orbit coupling is absent, and a stronger interface potential will induce a higher peak, as shown in figure 6(d). When the interface scattering potential goes to infinity, the peak goes to the quantized value, i.e. $2{{e}^{2}}/h$, and the width of the peak goes to zero. A quantized peak located at zero-bias voltage with vanishing width is apparently a manifestation of the Majorana end states.

Figure 7(a) shows that when the chemical potentials are not mismatched between the normal lead and the heterostructure superconductor, and the magnetic field is absent, there exists a sharp peak corresponding to a resonant spin-reversed Andreev reflection at the gap boundary, similar to the ${\rm N}-{\rm S}$ junction [53]. Note that in the absence of the magnetic field, there are no spin-reversed normal reflections and equal-spin Andreev reflections, even with strong spin–orbit coupling. This indicates that the magnetic field is a necessity to induce equal-spin pairing. As shown in figure 7(b), increasing the magnetic field will drive the peak left (zero-bias voltage), and the finite magnetic field also drives the peak away from the gap boundary. However, when there is a big mismatch between the chemicals, which is usually needed to guarantee that the magnetic field satisfying the topological criterion is still not large enough to break down the superconductivity, such an interesting phenomenon is absent (no finite-bias peak appears in figures 5(e) to (f)). Once the magnetic field reaches the critical value ${{B}_{c}}=\sqrt{\tilde{\mu }_{s}^{2}+\Delta _{s}^{2}}$, a zero-bias conductance peak is formed; however, the peak that is formed has a significantly reduced height as shown in figure 7(c). A sudden reduction of the peak height may imply that the zero-bias conductance peak and the finite-bias peak have different origins. For weak or intermediate spin–orbit coupling, further increasing the magnetic field has little effect on the zero-bias conductance peak; however, when the spin–orbit coupling is strong enough, the peak height monotonically increases to a parameter-dependent saturation value with an increasing magnetic field (here we do not consider the breakdown of superconductivity due to a strong magnetic field), as shown in figure 7(d). A further study in the stronger spin–orbit coupling region shows that the absence of a potential chemical mismatch makes the approach of zero-bias conductance peak to the quantized value even more difficult.

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To discuss the effects of the pairing potential on the tunneling potential, here we adopt the experimental parameters: $m=0.015\;{{m}_{e}}$, $m{{\alpha }^{2}}/2=\;50\;\mu$ eV, ${{\Delta }_{s}}=0.25$ meV, ${{\tilde{\mu }}_{\,s}}=0$ [39], and we choose ${{\mu }_{\,n}}=20$ meV. Figure 8(a) shows the tunneling spectroscopy at a temperature of zero. Comparing the zero-bias conductance peak with the experimentʼs measured value of $\sim 0.1{{e}^{2}}/h$, the result should still be several times larger, even if we consider the temperatureʼs smearing effect. However, as discussed previously, decreasing spin–orbit coupling (intermediate region) greatly reduces the peak height. In figure 8(b), it is shown that halving the spin–orbit coupling almost corresponds to halving the peak height. Therefore, if the spin–orbit coupling is several times smaller than the reported one, the height of the zero-bias conductance peak will decrease to be comparable with the experimentʼs measured value. Figure 8(b) also shows that by decreasing the pairing potential, the peak height is greatly increased. To achieve a sufficiently small-pairing potential, the zero-bias conductance peak is found to be almost quantized. This result seems counterintuitive, as the pairing potential is a necessity to induce the topological superconductor. This confusion can be clarified by the fact that when ${{\Delta }_{s}}\lt {{V}_{z}}\ll m{{\alpha }^{2}}/2$, the upper bandʼs effect is negligible, and as a result, the system becomes an 'effective p-wave' superconductor [13]. This suggests that to observe a more striking peak during experiments, it is better to choose a relatively weaker pairing potential proximity superconductor.

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The effects of finite length L for the ${\rm N}-{\rm hS}$ junction are similar to the ${\rm N}-p{\rm S}$ junction; that is, a conductance peak will locate at a finite-bias voltage when L is not sufficiently long. With the increase of L, the peak moves toward the zero-bias voltage; see figure 8(c). For a sufficiently long wire, we find that by increasing the magnetic field, the width of the zero-bias conductance peak will be greatly widened; see figure 8(d).