In recent years, Majorana bound states (MBSs), which are fundamentally solid state counterpart of Majorana fermions that are their own antiparticles in high-energy physics, have been extensively studied theoretically and experimentally.^[1,2] They are zero-energy quasiparticles realized at the edges or at topological defects of nontrivial superconductors, and are topologically protected.^[3–9] The topological protection, which is based on the separation of a pair of MBSs, results in the non-abelian statistics making MBSs ideal building blocks for topological quantum computation free from decoherence.^[10,11] With the rapid development of nano-fabrication techniques, topologically protected MBSs are now mainly realized from the spatially separated Andreev bound states (ABSs) in hybrid superconductor-semiconductor nanowires.^[12–18]

Since it is easy to control their electronic properties, the quantum dots (QDs),^[19–30] quantum rings,^[31] as well as Josephson junctions^[17,32–34] may be excellent platforms to detect the MBSs in solids. Experimentally, a series of quantum transport measurements for MBSs embedded in a semiconducting nanowire (with proximity-induced superconductivity) coupled to a QD have been performed. As an example, Deng et al.^[15,16] put forward a spectrometer comprised of a semiconducting nanowire with proximity-induced superconductivity for the degree of Majorana nonlocality in a QD by measuring the zero-bias conductance peak (ZBCP) under various conditions. However, in fact, the ZBCP can also originate from other physical mechanisms, such as the Kondo effect, disorder, and the ABSs in hybrid devices. To clarify the origin of the ZBCP observed in the quantum transport measurements, several theoretical schemes were proposed to analyze the tunneling spectroscopy of a QD which is coupled to both ends of the nanowire hosting the MBSs. For instance, Liu et al.^[26] theoretically considered the interplay between ABSs and MBSs in disorder-free QD nanowire semiconductor systems with proximity-induced superconductivity. Zhang et al. have successfully measured a quantized conductance plateau at 2e²/h in the zero-bias conductance in indium antimonide semiconductor nanowires covered with an aluminium superconducting shell, which supports the existence of Majorana zero-modes therein and may pave a way for the experiments that could lead to topological quantum computing.^[28,30] More recently, a definitive detection of the parity anomaly in a topological superconductor ring was also proposed.^[31] Furthermore, Floquet MBSs (zero mode and π mode) were theoretically studied in planar Josephson junctions, in which the subharmonic response to the external drive tuned by a top gate was found when both zero and π Majorana modes are present.^[34] Additionally, in the latest reports,^[35–38] nanosystems consisting of a hybrid superconducting nanowire coupled to a single QD and/or double quantum dots (DQDs) attached to a single normal metallic lead have been theoretically discussed.

Meanwhile, scanning tunneling microscopy (STM)^[39–42] measurements of MBSs in condensed matters have also been extensively carried out to study the electronic and spintronic properties of MBSs. Compared to the conventional quantum transport techniques, the STM has its own advantages in probing and even manipulating MBSs, from which the more precise quantum processes and mechanisms become possible to reveal. In particular, due to the nontrivial topological nature of MBSs, in situ designing of specific MBS configurations can be harnessed to image extraordinary quantum properties of MBSs. Recently, Sun et al. have probed the spin selective Andreev reflection of Majorana zero modes in a topological superconductor of the Bi₂Te₃/NbSe₂ heterostructures using a spin-polarized scanning tunneling microscope (STM). The spin dependent tunneling effect observed in their STM experiments provides direct evidence of Majorana zero modes (MZM).^[43] More recently, by employing a spin-polarized STM, Jeon et al. have also experimentally found that MBSs realized in self-assembled Fe chains on the surface of Pb have a spin polarization that exceeds the value stemming from the magnetism of these chains.^[44] By using an STM, a zero-bias peak inside a vortex core induced by MBSs has been observed on the superconducting Dirac surface state of the iron-based superconductor FeTe_0.55Se_0.45.^[45–47] Jäck and his collaborators have also performed the STM measurements for probing the MZM in a topologically protected edge of bismuth(111) films.^[48] Most recently, Chen et al. have reported their STM experimental finding of MZM simultaneously appearing at both ends of a one-dimensional atomic line defect in monolayer iron-based superconductor FeTe_0.5Se_0.5.^[49] Additionally, QD molecules have been successfully constructed on a semiconductor surface using atom manipulation by STM.^[50] On the theoretical side, the possibility to detect the spatial profile of Majorana fermions by using STM tips that are made of either a normal or a superconducting material have also been investigated, but no quantum dot is taken into account.^[51–54]

Motivated by both advanced quantum transport measurements for MBSs through QD coupled with topological superconducting nanowires^[15,16] and dramatic STM measurements of MBSs on the surfaces of topological superconductors,^[43,44] we would like to theoretically consider a model which is composed of an STM tip and DQDs coupled to the ends of nanowire where MBSs are induced due to the proximity effect of superconductivity, see Fig. 1. Differing from most of the previous quantum transport models reported in literature, in which QD usually couples to double (left and right) leads, in our present model we just consider a single normal metallic lead, i.e., the STM tip. Moreover, we emphasize that in our model, DQDs are taken into account, from which the spectral function of each QD may provide significant information about the effective tunneling (or not) of the MBSs into the QD. By comparing the spectral functions for the cases with and without coupling between the DQD and MBSs, two generic types of signatures of the MBSs presence in QD-1 or QD-2 may be identified.

Fig. 1.

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Fig. 1. Schematic plot of the hybrid STM-DQD-MBS interferometer. The system consists of a scanning tunneling STM tip (olive), DQDs (blue), and a semiconductor nanowire (purple) carrying MBSs (red) at its ends induced by a partially covered s-wave superconductor shell (green). The superconducting proximity in the nanowire (purple) is supposed to be realized by epitaxy Al[28] (green) on top of it, and then the MBSs are induced at the ends of nanowire (red points). The chemical potential in the segment covered with the superconducting shell may be tuned by applying voltages to the two long super-gates (yellow). The tunnel gates (orange steps) are employed to induce a tunnel barrier between the DQDs and the MBSs, and then tune the hybridizations λ1 and λ2 between the MBSs and DQDs. The coupling between MBSs η1 and η2 is denoted by εM. The energy levels of QD-1 and QD-2 are ε1 and ε2, respectively. The STM tip is coupled to DQDs with different coupling strengths V1 and V2, from which one could confirm the presence of MBSs and to probe information of MBSs. The interference contributions in the spectral functions of QDs are introduced by the electrons tunneling from QD-1 and QD-2 into the STM tip, respectively.

As shown in Fig. 1, DQDs are connected to the STM tip with different coupling strengths V₁ and V₂, and to the MBSs, which are denoted by η₁ and η₂, at the two ends of a semiconductor-superconductor nanowire with hybridizations λ₁ and λ₂, respectively. The MBS-QD coupling is local, and the wave functions of the MBSs at the locations where coupling happens are nearly identical. In Fig. 1, the superconducting proximity in the nanowire (purple) is supposed to be realized by epitaxy Al (see, for example, Ref. [28]) on top of it, and then the MBSs are induced at the ends of the nanowire (red points). Furthermore, the tunnel gates are covered on the tunnel junctions (purple steps) between MBSs and DQDs, and then the coupling strengths λ₁ and λ₂ could be tuned by the tunnel gates. In addition, the super gates are located at the sides of the nanowire with MBSs, and then the chemical potential in the segment covered by a superconducting shell may be tuned by the super gates. Under the resonance condition, i.e., the energy levels of DQDs are aligned to the chemical potential of the STM tip, the zero-energy spectral functions in the DQDs A₁₍₂₎(0) are 1/2 regardless of the value of the relative strength of the coupling strengths V₁ and V₂, and the difference between the energy levels of DQDs. This indicates the formation of the MBSs in the dots. In the presence of the overlapping between the MBSs in different dots, the property of A₁₍₂₎(ω) = 1/2 is retained in spite of the value of V₁/V₂, but it will be changed by the difference between the levels of DQDs.

The effective Hamiltonian of our model shown in Fig. 1 takes the following form:

Figure 2 shows the spectral functions under the conditions of γ₂ = γ₁, ε₁ = ε₂ = 0 (on-resonance) and ε_M = 0 for different values of λ₂. When QD-2 is decoupled from the MBSs (λ₂ = 0) as shown by the solid black lines in Fig. 2, the spectral functions at ω = 0 (corresponding to the case of the energy level of QD is aligned to the Fermi energy) take a δ-type function such that A_i(ω) = δ(ω-ε_i) with i = 1,2. There are also two broad peaks located around ω = ± λ₁, respectively. Now the electrons are bounded to the resonance state ω = ε_i and the information of the MBSs is overwhelmed. Turning on the coupling between QD-2 and the nanowire (λ₂ = 0.03), the spectral function A₁(ω) develops four peaks around ω = ± λ₁, ± λ₂ due to the energy level splitting by the interaction between the QD and the MBSs as indicated by the red line in Fig. 2(a). With further increasing λ₂, the height of two peaks around ω = ± λ₂ is lowered, and eventually they will overlap with those around ω = ± λ₁. For the case of λ₂ < λ₁, the peaks around ω = ± λ₁ are almost unchanged by the value of λ₂. For λ₂ ≥ λ₁, however, the position and height of them obviously depend on the value of λ₂. The spectral function A₂(ω) shows a double-peak configuration centered around ω = ± λ₂. Interestingly, the spectral function A₁₍₂₎(ω = 0) = 1/2 for any values of λ₂ > 0, indicating the existence of the MBSs.

Fig. 2.

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	Fig. 2. Spectral functions A1(ω) and A2(ω) in the QD-1 and QD-2 versus the spectral frequency ω for varying the coupling strength λ2 between QD-2 and MBSs. In the simulations, other parameters are chosen as γ1 = λ1 = 0.1, εM = ε1 = ε2 = 0.

In our STM-DQD-MBS model, the coupling strength between the STM tip and DQDs could be easily tuned via moving the position of the STM tip. If the STM tip is moved closer to the QD-1, then the coupling strength with QD-1 will be stronger, and the coupling strength with QD-2 will be relatively weaker. Therefore, it is necessary to study the influence of the coupling strength between the STM tip and DQDs on the spectral function. In Fig. 3 we show the impacts of the symmetry of the QD-STM coupling γ₂/γ₁ on the MBSs existing in the QD. The spectral function in QD-1 A₁(ω) shows a triple-peak configuration for γ₂ = 0 [see the black line in Fig. 3(a)]. With increasing γ₂, the two peaks around ω = ± λ₁ becomes more sharp and higher, and A₁(ω) changes into double-peak configuration as found in Fig. 2(a). It is found that the MBSs in QD-1 is unperturbed by the variation of γ₂, as indicated by the constant value of A₁(0) = 1/2. This is because the MBSs at the two ends of the nanowire are isolated for ε_M = 0, and the information of QD-2 cannot enter into the topologically protected zero-energy MBS in QD-1. The properties of A₂(ω) are essentially resembling to those of A₁(ω). Under the condition of γ₂ = 0, QD-2 is totally decoupled from the STM tip and the corresponding spectral function is zero [see the black line in Fig. 3(b)]. At weak γ₂, A₂(ω) shows a triple-peak configuration, which involves into a double-peak one. As long as γ₂ > 0, the spectral function at zero Fermi energy is fixed at A₂(0) = 1/2, indicating the stable existence of MBSs.

Fig. 3.

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	Fig. 3. Spectral functions A1(ω) and A2(ω) varying with ω for different values of coupling strength between the STM tip and DQ-2 γ2. In the calculations, we choose λ2 = 0.1, and other parameters are the same as those in Fig. 2.

The energy levels of DQDs are respectively considered as ε₁ and ε₂ in our model. In the above calculations of Figs. 2 and 3, ε₁ = ε₂ = 0 is chosen. In fact, the energy levels of DQDs may be different from each other, which could be easily tuned in experiments. Therefore, it is necessary to analyze influence of the difference between the energy levels of DQDs on the effective tunneling of the MBSs into the QD. Here, the difference between the energy levels of DQDs, i.e., levels’ detuning Δ is defined as ε₁( 2 ) = ε₀ + (–),Δ/2. Typical results of spectrum function A_i(ω) for several values of levels’ detuning Δ are shown in Fig. 4. For the case of λ₁ = 0.1 and λ₂ = 0.06 as shown in Fig. 4(a), A₁(ω) (thick lines) has four peaks under the condition of Δ = 0, which are symmetric with respective to the point of ω = 0, as found in Fig. 2(a). In the presence of the levels’ detuning, namely Δ ≠ 0, the peaks move toward lower energy regime with the reduction of the hight of peaks. The value of A₁(0), however, remains unchanged and the MBS bounded to QD-1 is quite stable against the level difference between the QD. Similar to the behavior of A₁(ω), the peaks in A₂(ω) (thin lines) move toward high energy region while the height of peaks are suppressed and the zero energy spectral function remains a half of unity, i.e., A₂(0) = 1/2. If λ₁ = λ₂ as shown in Fig. 4(b), A₁(ω) (thick lines) and A₂(ω) (thin lines) are mirror symmetric to ω = 0, and the characters of the peaks resemble those of in Fig. 4(a). As a consequence, Fig. 4 demonstrates that the topologically protected Majorana zero-energy mode in our STM-DQD-MBS interferemeter are insensitive to the variation of the energy levels of DQDs since electrons can hardly enter into these states. Moreover, we find that these MBSs formed in the DQDs are stable regardless of the relative strengths of λ₁ and λ₂.

Fig. 4.

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	Fig. 4. Spectral functions A1(ω) (thick lines) and A2(ω) (thin lines) versus ω for different levels detuning Δ between QD-1 and QD-2. In the calculations, λ2 = 0.06 for (a) and λ2 = 0.1 for (b) are chosen, and other parameters are the same as those in Fig. 2.

In all the above discussions, the overlap between the two MBSs is not considered (ε_M = 0), while in Fig. 5 we present the spectral functions in the presence of overlap between the two MBSs (ε_M ≠ 0). First of all, we find that the zero energy spectral functions remain numerically unchanged, i.e., A₁₍₂₎(0) = 1/2, regardless of the value of ε_M in both the asymmetric hybridization case of λ₁ > λ₂ [see Figs. 5(a) and 5(b)] and the symmetric hybridization case of λ₁ = λ₂ [see Fig. 5(c)]. For the general cases with various parameters, such as ε₁, ε₂, λ₁, λ₂, γ₁, γ₂, and ε_M shown in our model, it is difficult to get a compactly analytical expression of Green’s function of QDs in the presence of MBSs. However, for a special case of γ₁ = γ₂ = γ, λ₁ = λ₂ = λ, and ε₁ = ε₂ = 0, a compact expression of zero-energy Green’s function of QD G_d,i(ω → 0 ) may be approximated as

Fig. 5.

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	Fig. 5. Spectral functions varying with ω for different values of the coupling between the two MBSs εM. In the calculations, the hybridization between MBSs and QD-2 is chosen as λ2 = 0.06 in (a) and (b), and λ2 = 0.1 in (c). Other parameters are the same as those in Fig. 2.

In Figs. 6(a) and 6(b) we show the impact of different values of γ₂ on the spectral functions with a fixed overlap between MBSs. For instance, in Figs. 6(a) and 6(b) ε_M = 0.02 is chosen. Differing from Fig. 3 for the case of ε_M = 0, we can clearly find from Figs. 6(a) and 6(b) that the symmetry of the spectral functions is broken due to the nonzero overlap between MBSs. From Fig. 6(a), we observe that the four-peak configuration of A₁(ω) evolves a triple-peak one with increasing γ₂. This is because the interference effect between the DQDs via the STM tip becomes stronger. The peak around ω = –λ₂ is emerged into that of ω = λ₂, leading to the higher peak. With increasing γ₂, A₂(ω) develops a triple peaks whose positions are almost unchanged by the value of γ₂, whereas they become higher with increasing λ₂, see Fig. 6(b). It is a natural result that A₂(ω) = 0 when the coupling between the STM tip and QD-2 is absent (γ₂ = 0), see the black line in Fig. 6(b). It is not surprising that we find A_i(0) = 1/2 for any nonzero coupling strength γ_i = 1,2, which is consistent with the above discussions. Finally, in Fig. 6(c) we give the results of zero energy spectral functions A₁(ω = 0) and A₂(ω = 0) varying with energy level detuning Δ between DQDs for different overlap of MBSs ε_M. For the case of ε_M = 0, both A₁(ω) and A₂(ω) are equal to 1/2 regardless of the value of Δ. For the case of ε_M≠ 0, the spectral functions of DQDs equal to 1/2 only for Δ = 0, and are mirror-symmetric with respective to Δ = 0. They also develop peaks for nonzero ε_M.

Fig. 6.

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Fig. 6. (a) and (b) Spectral functions A1(ω) and A2(ω) varying with ω for different values of coupling strength between the STM tip and QD-2 γ2. (c) A1(ω) and A2(ω) as functions of levels detuning Δ for different overlap between MBSs εM. In all the calculations, symmetric coupling between STM and DQDs is considered by λ1 = λ2 = 0.1. In (a) and (b), the overlap between MBSs is chosen as εM = 0.06. Other parameters are the same as those in Fig. 2.

In all calculations of Figs. 2–6, we assume that the nanowire is driven into the topological phase that the MBSs exit at the ends of the nanowire. The MBSs at the end of the nanowire significantly influence the spectrum function of QD. A significant feature is that when MBSs and QDs are coupled with each other, no matter how other tunable parameters (such as coupling strength between the STM tip and QD, energy level detuning between QD-1 and QD-2, coupling between the two MBSs, and others) change within a reasonable range, the zero energy spectrum function of the QD always remains A_i(0) = 1/2. When MBSs and QDs are decoupled with each other, i.e., λ₁ = λ₂ = 0, the zero energy spectrum function of QD will be tuned by the coupling strength between the STM tip and the QD, the energy level detuning between QD-1 and QD-2, and other parameters. Especially, for the trivial case that λ₁ = λ₂ = 0, ε₁ = ε₂ = 0, and γ₁ = γ₂ = γ, the spectrum function of QD-i may be written as , and then . It is clear that a sharp jump in the zero energy spectrum function of QD by a factor of 1/2 is universal as long as the MBSs are taken into account and couple to the QD.^[36,37,55] Therefore, the spectrum function of QD, especially the zero energy spectrum function of QD, may be an evidence of the presence of MBSs, and then our model may provide a potential way to prob the MBSs.

Furthermore, our model could be easily generalized to a spin-polarized one. In other words, the STM tip could be assumed as an spin-polarized one. By applying an external magnetic field, the spin degeneceracy of energy levels of QD may be split via the Zeeman effect, and the spin of MBSs will be also polarized to a fixed direction (spin-up or spin-down). Thereby, the spin dependent tunneling effect and the spin blocked effect could be naturally taken into account in the spin-polarized model. As a result, the spin-resolved spectral function may be easily detected via a spin-polarized STM-DQD-MBS system, and the spin dependent effects therein may provide more interesting information and more direct evidence of Majorana zero modes in condensed matter. We will thoroughly study the spin polarized model in another work since it is beyond the scope of this work.

We have investigated the properties of the spectral functions in a DQD system coupled to a normal metallic STM tip and MBSs in a semiconductor-superconductor nanowires. It is found that the zero-energy spectral functions are always equal to half of unity, which indicates the formation of the MBSs in the QD and the zero-bias conductance peak. If the MBSs in DQDs are overlapped, the formation of the MBSs is favorable for the case of identical levels of QD. The presently advanced STM technology and quantum device fabrication technology enable our theoretical model to be easily implemented in experiments to validate our theoretical results. Therefore, we hope our theoretical findings in this work could be experimentally checked in the near future.