Majorana corner and hinge modes in second-order topological insulator-superconductor heterostructures

TL;DR

This paper explores second-order topological insulator/superconductor heterostructures, revealing they can host Majorana modes without special pairings or magnetic fields, advancing potential platforms for quantum computing.

Contribution

It introduces second-order topological insulator/superconductor heterostructures as natural realizations of second-order topological superconductors hosting Majorana modes.

Findings

Realization of Majorana corner modes in 2D heterostructures

Chiral Majorana hinge modes in 3D heterostructures

No need for special pairings or magnetic fields

Abstract

As platforms of Majorana modes, topological insulator (quantum anomalous Hall insulator)/superconductor (SC) heterostructures have attracted tremendous attention over the past decade. Here we substitute the topological insulator by its higher-order counterparts. Concretely, we consider second-order topological insulators (SOTIs) without time-reversal symmetry and investigate SOTI/SC heterostructures in both two and three dimensions. Remarkably, we find that such novel heterostructures provide natural realizations of second-order topological superconductors (SOTSCs) which host Majorana corner modes in two dimensions and chiral Majorana hinge modes in three dimensions. As here the realization of SOTSCs requires neither special pairings nor magnetic fields, such SOTI/SC heterostructures are outstanding platforms of Majorana modes and may have wide applications in future.

Equations

$$H(\mathbf{k})$$

$$H_0(-i{\partial }_x)$$

$$H_p(-i{\partial }_x,k_y)$$

$${\psi }_{\alpha }(x)={\mathcal{N}}_x\sin ({\eta }_{1}x)e^{-{\eta }_{2}x}{e}^{i{k}_{y}y}{\chi }_{\alpha },$$

$$H_{\mathrm{I},\alpha \beta }(k_y)={\int }_0^{+\infty }{\psi }_{\alpha }^{†}(x)(-i{\partial }_x,k_y){\psi }_{\beta }(x)dx.$$

$$H_{\mathrm{I}}(k_y)=-{\lambda }_y{k}_y{s}_z+M_{\mathrm{I},\mathrm{\Lambda }}{s}_y+M_{\mathrm{I},\mathrm{S}}{s}_y{\tau }_y,$$

$$M_{\mathrm{I},\mathrm{\Lambda }}$$

$$M_{\mathrm{I},\mathrm{S}}$$

$$H_{\mathrm{II}}(k_x)$$

$$H_{\mathrm{III}}(k_y)$$

$$H_{\mathrm{IV}}(k_x)$$

$$H_{\mathrm{Edge}}=-i\lambda (l){\partial }_l{s}_z+M_{\mathrm{\Lambda }}(l)s_y+M_{\mathrm{S}}(l)s_y{\tau }_y,$$

$$H_{{\tau }_y=1}$$

$$H_{{\tau }_y=-1}$$

$$H(\mathbf{k})$$

$$E(\mathbf{k})=\pm \sqrt{F(k)\pm 2\sqrt{G(k)}},$$

Full text

Over the past decade, topological superconductors (TSCs) have attracted continuous and tremendous attentionQi and Zhang (2011); Alicea (2012); Beenakker (2013); Stanescu and Tewari (2013); Leijnse and Flensberg (2012); Elliott and Franz (2015); Sarma et al. (2015); Sato and Fujimoto (2016); Aguado (2017). Among various TSCs, one-dimensional ($1d$) and two-dimensional ($2d$) TSCs without time-reversal symmetry (TRS) have attracted particular interest as they harbor Majorana zero modes (MZMs) at their boundariesKitaev (2001); Oreg et al. (2010); Lutchyn et al. (2010) and in the cores of vorticesRead and Green (2000); Fu and Kane (2008); Sau et al. (2010); Alicea (2010), respectively. Owing to their fractional nature, MZMs are ideal candidates to construct nonlocal qubits immune to local decoherenceKitaev (2001). Moreover, owing to their non-Abelian statisticsIvanov (2001), their braiding operations are found to realize elementary quantum gates. Thus, MZMs are believed to be building blocks of topological quantum computationNayak et al. (2008) and have been actively sought in experimentsMourik et al. (2012); Rokhinson et al. (2012); Deng et al. (2012); Das et al. (2012); Finck et al. (2013); Nadj-Perge et al. (2014); Albrecht et al. (2016); Deng et al. (2016); Zhang et al. (2018a); Sun et al. (2016); Wang et al. (2018).

As is known, odd-parity superconductors (SCs) provide natural realizations of TSCs, however, they are unfortunately rare in nature. In a seminal paperFu and Kane (2008), Fu and Kane pointed out that topological insulator (TI)/SC heterostructures provide an effective realization of odd-parity superconductivity. Accordingly, in the presence of magnetic field, vortices emerging in such heterostructures are found to carry MZMs. In a later influential paperQi et al. (2010), Qi et al pointed out that quantum anomalous Hall insulator (QAHI)/SC heterostructures provide a simple realization of $2d$ chiral TSCs which harbor not only vortex-core MZMs, but also chiral Majorana edge modes. These two theoretical works have triggered a lot of experimental works on TI(QAHI)/SC heterostructuresSun et al. (2016); Wang et al. (2018, 2012, 2013); Zareapour et al. (2012); Xu et al. (2015); Lv et al. (2016); Zhang et al. (2018b); Liu et al. (2018a); He et al. (2017); Kayyalha et al. (2019); Chen et al. (2019); Zhu et al. (2019), and remarkable progress in detecting vortex-core MZMs has been witnessed in recent yearsSun et al. (2016); Wang et al. (2018); Chen et al. (2019); Zhu et al. (2019).

Very recently, TIs and TSCs have been generalized to include their higher-order counterpartsBenalcazar et al. (2017a); Song et al. (2017); Langbehn et al. (2017); Benalcazar et al. (2017b); Schindler et al. (2018a); Ezawa (2018); Rasmussen and Lu (2018); You et al. (2018); Khalaf (2018); Geier et al. (2018); Franca et al. (2018); Călugăru et al. (2019); Trifunovic and Brouwer (2019); Ahn and Yang (2018); Kudo et al. (2019). Importantly, higher-order TIs (HOTIs) and TSCs (HOTSCs) have extended the conventional bulk-boundary correspondence. Accordingly, an $n$-th order TI or TSC in $d$ dimensions host ($d-n$)-dimensional boundary modes. For instance, a second-order TI (SOTI) in $2d$ and $3d$ host zero-dimensional ($0d$) corner modes and $1d$ hinge modes, respectively. The existence of HOTIs and the lessons from the study of TI(QAHI)/SC heterostructures lead us to ask the natural question that whether Majorana corner modes (MCMs, i.e., MZMs bound at the corners) or chiral Majorana hinge modes (CMHMs) can also be achieved in a HOTI/SC heterostructure. It is worth noting that such a question is quite timely as recently the electronic material candidates for SOTIs, both in two dimensions ($2D$) and three dimensions ($3D$), are growingSchindler et al. (2018b); Yue et al. (2019); Wang et al. (2018); Xu et al. (2019); Sheng et al. (2019); Lee et al. (2019); Chen et al. (2019). Moreover, signature of MZM has also been observed in a heterostructure which consists of a bismuth thin film (a SOTI with TRSSchindler et al. (2018b)), a conventional $s$-wave SC, and magnetic iron clustersJäck et al. (2019).

In this work, we consider SOTIs without TRS and investigate SOTI/SC heterostructures in both $2D$ and $3D$. Remarkably, we find that such heterostructures provide natural realizations of second-order topological superconductors (SOTSCs) which host MCMs in $2D$ and CMHMs in $3D$. Furthermore, here the realization of SOTSCs does not require the pairing of SCs to take any specific form. It can be achieved for both unconventional SCs and conventional $s$-wave SCs. In addition, it does not need magnetic fields or the deposition of magnetic atoms. In comparison to previous proposalsYan et al. (2017); Shapourian et al. (2018); Zhu (2018); Yan et al. (2018); Wang et al. (2018a, b); Liu et al. (2018b); Hsu et al. (2018); Pan et al. (2018); Bultinck et al. (2019); Peng and Xu (2019); Volpez et al. (2019); Wu et al. (2019); Zeng et al. (2019); Ghorashi et al. (2019); Zhang et al. (2019); Kheirkhah et al. (2019); Hsu et al. (2019); Wu et al. (2019a); Laubscher et al. (2019); Zhang and Trauzettel (2019); Zhu (2019); Zhang et al. (2019); Wu et al. (2019b); Yan (2019); Ahn and Yang (2019), these merits make SOTI/SC heterostructures stand out, and potentially allow them to have wide applications in topological quantum computationYou et al. (2018); Weda Bomantara and Gong (2019); Pahomi et al. (2019).

MCMs in a $2d$ SOTI-SC heterostructure.— A $2d$ SOTI/SC heterostructure (Fig.1) could be described by a Bogoliubov-de Gennes (BdG) Hamiltonian, $H=\sum_{{\bf k}}\Psi_{{\bf k}}^{{\dagger}}H({\bf k})\Psi_{{\bf k}}$, with $\Psi_{{\bf k}}=(c_{a,{\bf k}\uparrow},c_{b,{\bf k}\uparrow},c_{a,{\bf k}\downarrow},c_{b,{\bf k}\downarrow},c_{a,-{\bf k}\uparrow}^{{\dagger}},c_{b,-{\bf k}\uparrow}^{{\dagger}},c_{a,-{\bf k}\downarrow}^{{\dagger}},c_{b,-{\bf k}\downarrow}^{{\dagger}})^{T}$ and

[TABLE]

where $\sigma_{i}$, $s_{i}$ and $\tau_{i}$ are Pauli matrices in orbit $(a,b)$, spin ($\uparrow,\downarrow$) and particle-hole spaces, respectively; $\epsilon({\bf k})=m_{0}-t_{x}\cos k_{x}-t_{y}\cos k_{y}$ is the kinetic energy; $\Lambda({\bf k})=\Lambda_{x}\cos k_{x}-\Lambda_{y}\cos k_{y}$ is a TRS breaking term crucial for the realization of SOTI; $\mu$ is the chemical potential, and $\Delta({\bf k})=\Delta_{0}+\Delta_{x}\cos k_{x}+\Delta_{y}\cos k_{y}$ represents the pairing. Such a form is general enough to model $s$-wave, $s_{\pm}$-wave and $d$-wave pairingsYan et al. (2018). For convenience, the lattice constants have been set to unit, and $t_{x,y}$, $\lambda_{x,y}$ and $\Lambda_{x,y}$ are set to be positive throughout this work.

Let us focus on the normal state first. Without the terms in the second line of Eq.(1), the Hamiltonian describes a $2d$ first-order TI when $\prod_{\alpha,\beta=\pm 1}(m_{0}+\alpha t_{x}+\beta t_{y})<0$Fu and Kane (2007). Accordingly, when open boundary condition is taken, gapless helical modes will appear on the boundary. Adding the $\Lambda({\bf k})$ term breaks TRS and consequently gaps out the helical modes, resulting in a transition from a first-order TI to a SOTI. When open boundary conditions are taken in both the $x$ and $y$ directions, one can find that in the SOTI phase, each corner of the system will harbor one zero-energy bound state with a fractional charge $e/2$Song et al. (2017). The pinning of the corner modes’ energy to zero is due to the existence of a chiral symmetry (the operator is $\sigma_{x}s_{y}$. When superconductivity enters, the operator is accordingly modified as $\sigma_{x}s_{y}\tau_{z}$). However, this chiral symmetry is just an accidental symmetry, adding an arbitrary term proportional to the identity matrix (e.g., the chemical potential) immediately breaks this symmetry and accordingly shifts the energy away from zero. Nevertheless, whether the chiral symmetry is preserved or not does not affect our following discussions since the particle-hole symmetry of a SC is sufficient to guarantee the topological robustness of MCMs.

To see the effect of superconductivity intuitively, let us focus on the case with chiral symmetry first. As is known, when a chiral electronic mode is in proximity to a SC, it becomes two chiral Majorana modes in the weak-pairing limitQi et al. (2010). Similarly, when a SOTI is in proximity to a SC, each charged corner mode will become two MCMs. However, as the wave functions of the two MCMs overlap in space, they are not robust against local perturbations and disorders. Therefore, one may naively think that robust MZMs in general can not be realized in a SOTI/SC heterostructure. However, the simple “one-to-two” picture above is only valid in the weak-pairing limit. In Fig.2(a) , we have shown explicitly that for a sample with square geometry, when the pairing amplitude exceeds some critical value, a SOTSC phase with four well-separated MCMs can be achieved even though the pairing of SC is $s$-wave. As now each corner has only one MZM, the particle-hole symmetry guarantees that these MCMs are robust against local perturbations, as well as random disorders if the disorder strength is weaker than some critical value.

Edge theory.— To see how the SOTSC phase and the concomitant MCMs are realized, we investigate the edge theory for an intuitive understanding. For simplicity, we still focus on the $\mu=0$ case and consider the continuum model corresponding to a low-energy expansion of the lattice Hamiltonian in Eq. (1) to second order around ${\bf k}=(0,0)$:

[TABLE]

where $m=t_{x}+t_{y}-m_{0}$, $\Lambda(k_{x},k_{y})=\Lambda-\frac{1}{2}(\Lambda_{x}k_{x}^{2}-\Lambda_{y}k_{y}^{2})$ with $\Lambda=\Lambda_{x}-\Lambda_{y}$, and $\Delta(k_{x},k_{y})=\Delta-\frac{1}{2}(\Delta_{x}k_{x}^{2}+\Delta_{y}k_{y}^{2})$ with $\Delta=\Delta_{0}+\Delta_{x}+\Delta_{y}$. We consider $m>0$ so that without the terms in the second line the Hamiltonian describes a first-order TI. For convenience, we label the four edges of a square sample I, II, III, and IV (see Fig. 1) and define a $1d$ “boundary coordinate” $l$ which grows in a counterclockwise fashion along the edges. Let us focus on the edge (I) first. To obtain the corresponding low-energy Hamiltonian which describes the edge modes, we perform the replacement $k_{x}\rightarrow-i\partial_{x}$ and decompose the Hamiltonian into two partsYan et al. (2018), i.e., $H=H_{0}+H_{p}$ with

[TABLE]

where $\Lambda(-i\partial_{x})=\Lambda+\frac{1}{2}\Lambda_{x}\partial_{x}^{2}$ and $\Delta(-i\partial_{x})=\Delta+\frac{1}{2}\Delta_{x}\partial_{x}^{2}$. Here we have neglected the insignificant $k_{y}^{2}$ terms for simplicity. Treating $H_{p}$ as a perturbation and first solving the eigenvalue problem $H_{0}(-i\partial_{x})\psi_{\alpha}(x)=E_{\alpha}\psi_{\alpha}(x)$ under the boundary condition $\psi_{\alpha}(0)=\psi_{\alpha}(+\infty)=0$, one can find that there are four zero-energy solutions, which read

[TABLE]

where $\mathcal{N}_{x}=2\sqrt{|\eta_{2}(\eta_{1}^{2}+\eta_{2}^{2})/\eta_{1}^{2}|}$ denotes the normalization factor, $\eta_{1}=\sqrt{2m/t_{x}-\lambda_{x}^{2}/t_{x}^{2}}$ and $\eta_{2}=\lambda_{x}/t_{x}$; The four spinors $\chi_{\alpha}$ are determined by $\sigma_{y}s_{z}\tau_{z}\chi_{\alpha}=-\chi_{\alpha}$. For their concrete forms, here we follow ref.Yan et al. (2018). Accordingly, the matrix elements of $H_{p}$ under the basis composed by the four zero-energy solutions are

[TABLE]

The corresponding low-energy Hamiltonian for edge (I) is

[TABLE]

where the two Dirac masses $M_{{\rm I,\Lambda}}$ and $M_{{\rm I,S}}$ are of different origins, and they are given by

[TABLE]

Similarly, the low-energy Hamiltonians for the other three edges are

[TABLE]

with $M_{{\rm II,\Lambda}}=M_{{\rm IV,\Lambda}}=-\Lambda-m\Lambda_{y}/t_{y}$, $M_{{\rm II,S}}=M_{{\rm IV,S}}=\Delta-m\Delta_{y}/t_{y}$, and $M_{{\rm III,\Lambda}}=M_{{\rm I,\Lambda}}$, $M_{{\rm III,S}}=M_{{\rm I,S}}$. By using the boundary coordinate, the low-energy Hamiltonian can be written compactly as

[TABLE]

where $\lambda(l)$, $M_{\rm\Lambda}(l)$ and $M_{{\rm S}}(l)$ are step functions with their values following the sequences: $\lambda(l)=\lambda_{x}$, $\lambda_{y}$, $\lambda_{x}$, $\lambda_{y}$, $M_{\rm\Lambda}(l)=-\Lambda+m\Lambda_{x}/t_{x}$, $-\Lambda-m\Lambda_{y}/t_{y}$, $-\Lambda+m\Lambda_{x}/t_{x}$, $-\Lambda-m\Lambda_{y}/t_{y}$, and $M_{\rm S}(l)=\Delta-m\Delta_{x}/t_{x}$, $\Delta-m\Delta_{y}/t_{y}$, $\Delta-m\Delta_{x}/t_{x}$, $\Delta-m\Delta_{y}/t_{y}$ for (I), (II), (III) and (IV), respectively.

Without loss of generality, let us focus on the case with $\Lambda_{x}=\Lambda_{y}$ so that $\Lambda=0$. In the absence of pairing, i.e., $M_{{\rm S}}(l)=0$, $H_{\rm Edge}$ reduces to a $2\times 2$ matrix. At each corner, $\lambda(l)$ does not change sign, but $M_{\rm\Lambda}(l)$ does, realizing a domain wall of Dirac mass which harbors one charged zero mode according to the Jackiw-Rebbi theoryJackiw and Rebbi (1976). When superconductivity enters, one can see that $H_{\rm Edge}$ is the direct sum of two independent parts, i.e., $H_{\rm Edge}=H_{\tau_{y}=1}\oplus H_{\tau_{y}=-1}$ with

[TABLE]

One can see that the Dirac mass induced by superconductivity takes different signs in the two parts. In the weak-pairing limit, $|M_{{\rm S}}(l)|<<|M_{\rm\Lambda}(l)|$, each part realizes one zero mode per corner. As the particle component and the hole component of these zero modes’ wave functions are equal (note $\tau_{y}\psi_{0}(l)=\pm\psi_{0}(l)$, where $\psi_{0}(l)$ denotes the wave function of zero mode), they are MZMs, agreeing with our previous argument that weak superconductivity will transform one charged zero mode to two MZMs. As now each corner harbors two MZMs, these MCMs are not stable. Indeed, we find that any finite $\mu$ or on-site potential will make them couple (the chemical potential term contains $\tau_{z}$, so it makes the $\tau_{y}=1$ part couple with the $\tau_{y}=-1$ part) and consequently destroy their self-conjugate nature. This can also be understood from the perspective that because $\mu$ shifts the energy of charged corner modes away from zero, the energy of corner modes will keep taking finite values if the superconductivity is very weak. Therefore, for the square geometry presented in Fig.1, robust MCMs are absent in the weak-pairing limit. Noteworthily, as $M_{\rm\Lambda}(l)$ is in fact sensitive to the orientation of edge, here we have emphasized the particular square geometry shown in Fig.1. As will see shortly, if the sample’s geometry is appropriately designed, the critical value of pairing amplitude for realizing robust MCMs can be very small, so even weak superconductivity is sufficient.

To see how robust MCMs emerge in a square sample, we take $s$-wave pairing for illustration (other more exotic cases can similarly be analyzed). Accordingly, $M_{\rm S}(l)=\Delta_{0}$ is uniform on the boundary. Without loss of generality, we further assume $m\Lambda_{y}/t_{y}>m\Lambda_{x}/t_{x}$. According to Eq.(10), one can find when $m\Lambda_{x}/t_{x}<\Delta_{0}<m\Lambda_{y}/t_{y}$, while the domain walls for $H_{\tau_{y}=1}$ are preserved since $m\Lambda_{x}/t_{x}+\Delta_{0}$ and $-m\Lambda_{y}/t_{y}+\Delta_{0}$ still take opposite signs, the ones for $H_{\tau_{y}=-1}$ are removed since now $m\Lambda_{x}/t_{x}-\Delta_{0}$ and $-m\Lambda_{y}/t_{y}-\Delta_{0}$ take same sign. As a result, there is only one MZM per corner in this regime, as shown in Fig.2(a). We have numerically checked that these MCMs are robust against local perturbations, doping and random disorders as long as the doping level and disorder strength are small than some critical values (note in Fig.2(a), $\mu=0.1$).

According to the criterion $m\Lambda_{x}/t_{x}<\Delta_{0}<m\Lambda_{y}/t_{y}$, one may make the conclusion that if the underlying pairing is $s$-wave, anisotropy is necessary for the realization of SOTSC. That is, if $\Lambda_{x}=\Lambda_{y}$, $t_{x}\neq t_{y}$ must be satisfied. However, anisotropy is in fact unnecessary. For the isotropic case with $t_{x}=t_{y}$ and $\Lambda_{x}=\Lambda_{y}$, $M_{\rm\Lambda}(l)$ follows the angle dependence $M_{\rm\Lambda}(l)\simeq m\Lambda_{x}\cos 2\theta/t_{x}$, where $\theta$ represents the angle relative to edge (I). This indicates that on the edge whose orientation is pointing to $\theta=\pi/4$, $M_{\rm\Lambda}(l)=0$. As a result, one can find that for the $\pi/4$-angle corner formed by edge (I) and the $\theta=\pi/4$-orientation edge, it will harbor one MZM as long as $0<|\Delta_{0}|<m\Lambda_{x}/t_{x}$. We demonstrate the validity of this analysis numerically, as shown in Figs.2(b)(c). According to the phase diagram in Fig.2(c), one can see that for an isosceles-right-triangle geometry, MCMs can exist for a quite broad range of $\mu$ and for infinitely weak pairing amplitudesup .

As for a SOTI, $M_{\rm\Lambda}(l)$ inevitably vanishes along some direction, this implies that a judicious design of the corners is always able to realize MCMs even though the superconductivity is weak. Clearly, this conclusion also holds for other unconventional SCs.

CMHMs in a $3d$ SOTI/SC heterostructure.— The scenario above can straightforwardly be generalized to $3D$. For example, if we have a $3d$ SOTI at hand, we can grow a thin film of $s$-wave SC on its surface (see Figs.3(a)(b)). Accordingly, the system could be modeled by $H=\sum_{{\bf k}}\Psi_{{\bf k}}^{{\dagger}}H({\bf k})\Psi_{{\bf k}}$ with

[TABLE]

where $\xi({\bf k})=m_{0}-t_{x}\cos k_{x}-t_{y}\cos k_{y}-t_{z}\cos k_{z}$. Without the terms in the second line, the Hamiltonian describes a strong TI when $\prod_{\alpha,\beta,\gamma=\pm 1}(m_{0}+\alpha t_{x}+\beta t_{y}+\gamma t_{z})<0$. Accordingly, when open boundary condition is taken, gapless Dirac surface states will appear on the boundary. The presence of the term $\Lambda({\bf k})$ gaps out the Dirac surface states on the four lateral surfaces (in $3D$, we take open boundary condition in both the $x$ and $y$ directions, and periodic boundary condition in the $z$ direction) and leaves one chiral electronic mode per hingeSchindler et al. (2018a).

As mentioned before, when superconductivity enters, one chiral electronic mode becomes two chiral Majorana modes in the weak-pairing limitQi et al. (2010). Unlike the MCMs in two dimensions, while here the wave functions of the two chiral Majorana modes also overlap in space, they are stable against perturbations since they are chiral in nature. Therefore, in the weak-pairing regime, there are two robust chiral Majorana modes per hinge (see Figs.3(a)(c)). Interestingly, we find that with the increase of pairing amplitude, a topological phase transition will take place on the boundary and accordingly a new SOTSC which host one robust chiral Majorana mode per hinge will be realized (see Figs.3(b)(d)). It is worth noting that when doing the calculation of the energy spectra presented in Figs.3(c)(d), the superconductivity has been taken to be uniform throughout the whole sample. It is apparent that this assumption is unrealistic for the heterostructure since deep in the bulk the superconductivity induced by proximity effect should vanish, however, the low-energy physics within the gap can be well captured since the in-gap states are located on the surfaces which are well in contact with the SC. In other words, here the derivation from real situation only has strong impact on the bulk states. In fact, if we focus on the in-gap states, we can also adopt the edge theory as in $2D$. For the geometry shown in Figs.3(a)(b), one can easily find that the criterion for realizing the SOTSC phase with one robust chiral Majorana mode per hinge is also $m\Lambda_{x}/t_{x}<\Delta_{0}<m\Lambda_{y}/t_{y}$ ($m=t_{x}+t_{y}+t_{z}-m_{0}>0$, $\mu=0$, and $\Lambda_{x}=\Lambda_{y}$ are also presumed). One can see that the results presented in Figs.3(c)(d) are consistent with this criterion. Similar to the $2d$ situation, the critical pairing amplitude can also be tuned to take a very small value if the sample’s geometry is appropriately designed.

Conclusions.— We have shown that SOTI/SC heterostructures provide promising new platforms of MCMs and CMHMs. As our proposed scheme requires neither special pairings nor magnetic fields, we believe it should be simple to implement experimentally. Consider the fast growth of material candidates for SOTIsSchindler et al. (2018b); Yue et al. (2019); Wang et al. (2018); Xu et al. (2019); Sheng et al. (2019); Lee et al. (2019); Chen et al. (2019), we can foresee that such novel heterostructures will be synthesised and investigated in the near future. Experimentally, MCMs and CMHMs can be probed by STM techniquesJäck et al. (2019) and transport experimentsGray et al. (2019).

Acknowlegements.— We would like to acknowledge the support by a startup grant at Sun Yat-sen University.