Introduction

A Majorana fermion (MF) is an anti-particle of itself, originally proposed as a candidate for neutrino1. Recently it was proposed that MF may be generated in some particular condensed matter systems. For example, a zero energy MF mode can appear in the vortex core of the s-wave superconductor/bulk topological insulator heterostructure2. As a variant, MF may also appear at the open ends of a semiconducting nano-ribbon with strong spin-orbital coupling, in proximity to an s-wave superconductor and under a high magnetic field3. Apart from such hetero-structures, MF may appear in vortices of pure systems, such as the superfluid 3He-B4 and the potential px + ipy-wave superconductor Sr2RuO45. The renewed extensive interests in search of MFs are mainly driven by the potential application of MFs. Since the MFs obey non-Abelian statistics, they can be utilized to perform fault-tolerant topological quantum computing6. In addition, the MF mode also leads a plethora of new intriguing effects, such as fractional Josephson effect7,8, resonant Andreev reflection9,10 and resonant crossed Andreev reflection11,12.

Recently important experimental progresses have been made for the topological insulator/s-wave superconductor heterostructure13,14. In the earlier theoretical proposal2, a thin layer of superconductor (SC) is to be deposited on a bulk topological insulator (TI). However, experimentally it is difficult to prepare a pristine bulk TI (such as Bi2Se3), while it is much easier to prepare a bulk SC (such as Ne2Se3). Therefore, in Ref. 13 a few (quintuple) layers of TI was deposited on the bulk SC substrate, resulting in a TI/SC configuration. Signatures of both superconductivity and the surface states of TI (away from the Fermi level) were observed13. More recently, the structure is further improved by replacing Bi2Se3 with Bi2Te314. Scanning tunneling microscopy measurements (STM) of the local density of states (LDOS) on the top layer revealed that the zero-bias peak in the LDOS is perfectly symmetric and does not branch in energy up to a finite distance away from the vortex core, if the number of TI layers n ≥ 4. This non-branching behavior is in contrast to the case of n < 4, in which the branching begins right away from the vortex core as in usual s-wave superconductors. The non-branching behavior was argued as the signature of MF. The experiment appears very promising. However, in order for the heterostructure to be applied, a few important issues need to be addressed. First, since the zero-bias-peak appears for any case of n in the experiment, it is necessary to identify/falsify the zero energy bound state as a MF, in addition to the identification based on the non-branching behavior. Second, it is necessary to determine and understand the optimal number of TI layers to realize MF. Finally, in a more general respect, there are inevitably imperfections or impurities in real samples. For applications it is important to understand how impurities would destroy a MF mode. We should mention that according to the experiment, the Fermi level is away from the Dirac point of the TI surface states (in fact close but below the conduction band). The situation is very different to the SC/TI/SC configuration with the Fermi level right at the Dirac point addressed elsewhere15,16,17.

Here we address the above issues theoretically. First, we demonstrate that MF modes appear on both top and bottom layers of TI and are well separated for a moderate n. The top MF becomes more extended with increasing n, in agrement with the experiment. Second, we show both analytically and numerically that right at the vortex core the MF mode is always accompanied by another low energy bound state, leading to a zero-bias peak plus a side peak in the local density of states (LDOS) therein. However, a local scalar impurity at the core can wipe out the accompanying side-peak state while leaving the zero-energy MF mode intact. Consequently the LDOS becomes symmetric at the vortex core and the peak does not branch near the vortex core, in agreement with the experiment. Finally but unfortunately, while the MF is extremely stable against a single local impurity, the stability in terms of the critical impurity strength is reduced drastically for a moderate concentration of impurities.

The 3d heterostructure

The in-plane structure of the experimental TI is a triangle lattice and each unit cell contains a quintuple layer in the out-of-plane direction. For convenience but without loss of generality, we model the 3D TI by a simple cubic lattice, with the hamiltonian

Here ψr is a four-spinor annihilation field at site r = (x, y, z), is a bond vector connecting nearest-neighbor sites and tb is the hopping parameter along bond b. On the other hand, Γ0 = η3 ⊗ σ0 and Γb = η1 ⊗ σ b are four Dirac matrices. The Pauli matrices η (σ) act on orbital (spin) basis. The parameters m controls the topology of the insulator and finally µ is the chemical potential. We take as the direction normal to the quintuple layer in Bi2Se3 or Bi2Te3 and we model the anisotropy by taking .

We assume that the SC substrate contacting the bottom layer leads to an effective s-wave pairing hamiltonian

Under an applied magnetic field along , vortices appear and thread the heterostructure. Experimentally the vortices are dilute for a magnetic field of the order of Tesla. Thus we consider an isolated vortex line in an L × L × n cubic lattice, as schematically shown in Fig. 1(a). The pairing potential is written as,

where Δ0 is the pairing amplitude in SC, r|| = (x, y) is the planar position vector (the origin of which is set at the center of the L × L lattice) with azimuthal angle , ξ is the coherence length of the SC substrate and finally f(z) is a proximity function. The concrete form of f(z) is unknown. However it is reasonable to take f(n) = Δn/Δ0, where Δn is the pairing gap measured at the top layer of a heterostructure with n layers of TI. In qualitative accordance to the experiment13,14, we take such a function as plot in Fig. 1(b). We assume ξ = ξ0/f(z), where ξ0 is the coherence length in the SC substrate, to reflect the fact that the effective pairing gap varies along z. In the London limit the magnetic field leads to a vector potential a(r) = π(−y, x, 0)/(2L2) in dimensionless units. This amounts to a unit of flux ϕ0 = hc/2e threading the planar area. The vector potential couples to electrons minimally via

in the first line of Eq.(1).

Figure 1
figure 1

(a) Schematic plot of a vortex threading the L × L × n TI above the SC substrate (gray box). (b) The assumed proximity function f(z).

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The complete hamiltonian is H = H0 + Hp. Formally we rewrite it in the Nambu space as

with the spinor and the corresponding single-particle hamiltonian h(r, r′) (each being an 8 × 8 matrix acting on spin, orbital and Nambu bases). It is easy to show that h satisfies the anti-commutation relation {τ1K, h} = 0, where τ1 is the first Pauli matrix acting in the Nambu space and K is the complex conjugation operator. By this antisymmetry, if h|Ψ〉 = E|Ψ〉, it is guaranteed that hτ1K|Ψ〉 = −Eτ1K|Ψ〉. Thus |Ψ〉 and τ1K|Ψ〉 form a pair of (orthogonal) states with opposite energies. Because of the doubling of degrees of freedom in the Nambu space, only one of such two states are independent. If E = 0, the two states become degenerate and any linear combination of them is also an eigenstate. In a particular combination resulting in a pair of new states |Ψ1,2〉, it is possible to enforce τ1K|Ψ1〉 = |Ψ1〉 and τ1K|Ψ2〉 = |Ψ2〉 (up to trivial phases), in addition to the orthogonality condition 〈Ψ1|Ψ2〉 = 0. The quasi-particle field operators associated with these states are self-hermitian, defining the MF modes. They would be stable against perturbations provided they are spatially bounded and not overlapping each other. Such a MF mode was argued to appear in the vortex core in a heterostructure with a layer of s-wave SC on a bulk TI2. The situation we are addressing is upside down in structure: n layers of TI on top of a bulk s-wave SC. For sufficiently large n, helical states appear on the surfaces of the otherwise isolated TI. By coupling to SC, there should be a MF mode near the interface (the bottom layer of TI) according to Ref. 2. For the same reason another MF mode should appear on the top layer if the proximity induced pairing gap persists over there. For an intermediate n, the postulated MF's near the bottom and top layers may couple to each other, lifting the energy degeneracy.

To proceed we investigate numerically the evolution of the lowest-lying state |Ψ0〉 (with energy E0 closest to zero) versus n to gain insights into how MF states would emerge out of |Ψ0〉 and τ1K|Ψ0〉. We define the probability density function

to reveal the detailed spatial profile of the emerging MF wavefunction(s) in the heterostructure. Here the trace is taken over spin, orbital and Nambu degrees of freedom. We notice that p0(r) is identical for both |Ψ0〉 and τ1K|Ψ0〉, although they are necessarily orthogonal if E0 ≠ 0. In the calculations we fix the parameters (t||, t⊥, m, µ, Δ0, ξ0) = (1, 0.5, 0.3, 0.23, 0.1, 10) without lost of generality. The smaller hopping t⊥ takes qualitative account of the anisotropy in real samples. Fig. 2 shows the resulting p0(r) (color scale) for n = 2, 4, 6, 8 and L = 150 = 15ξ0. We see that the state is bounded to the vortex line in all cases. It is fully connected across the layers for n = 2 in Fig. 2(a). In Fig. 2(b), p0(r) is mainly concentrated near the top and bottom layers, implying that this is not the usual Andreev bound state, but rather two MF states slightly reconstructed due to the coupling in the interior of the sample. Indeed, a separate calculation shows that in an isolated n ≥ 4 sample, gapless helical surface states begin to show up. Such states coupled to the pairing potential would lead to MF states. For larger n in Fig. 2(c) and (d), the function p0(r) becomes more disconnected along z due to weaker coupling between the top and bottom layers and at the same time more extended on the top layer where the effective pairing gap decreases for increasing n. Since an extended MF mode would be fragile, the above results suggest that there is an optimal value of n, say nopt, to look for MF mode on the top layer of the present heterostructure. Since the energy E0 is nonzero in our finite-sized sample, we check the scaling of E0 versus 1/L in Fig. 3. Clearly E0 is finite and asymptotically independent of L in the limit of large L for n = 2, while E0 → 0 with increasing L for n = 6. The case for n = 4 appears marginal due to the coupling between top and bottom layers. Thus we expect a genuine MF mode on the top layer for n ≥ nopt = 6 in the thermodynamic limit of L, supporting the experimental setup in search of MF. Of course the above nopt should only be taken qualitatively, given the difficulty to extract the exact parameters in the experiment.

Figure 2
figure 2

The wavefunction profile p0(r) for (a) n = 2, (b) n = 4, (c) n = 6 and (d) n = 8.

In each panel, a corner is cutoff to reveal the interior of the TI.

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Figure 3
figure 3

Finite size scaling of E0 versus 1/L for various n.

Symbols are the data and lines are guides to the eye.

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Before closing this section we remark that since the top and bottom MF's are bound to the vortex line, braiding two vortices leads to braiding of the MF pairs. The statistics during such a braiding is identical to that for conventional fermions. Thus in the present heterostructure it might be difficult to achieve non-Abelian statistics that requires braiding individual MF's.

Effective 2d models

In the experiment the LDOS is measured by STM. Since the contribution mainly comes from the top layer, it is beneficial to investigate the top layer in more details. To reduce the computation load, we resort to an effective 2D model describing the low-lying quasi-particle states. We first gain understanding by analytical analysis of the a continuum model. We then perform numerical simulations in a lattice model. We also study the robustness of the MF state against impurities and compare the LDOS to the unexpectedly symmetric zero bias peak observed in the experiment.

Analytical treatment in the continuum limit: In the low energy window, the effective hamiltonian in the Nambu space can be written as,

Here we used a special convention for the annihilation field and the group velocity vD of the helical states is set to unity. Henceforth we use r to denote the 2D position. We performed a spin rotation to write ∇ · σ instead of ∂xσy − ∂yσx · Δ1 (Δ2) is the real (imaginary) part of the s-wave pairing potential

where Δ0(r) is the pairing scale, ξ the coherence length and ϕ the azimuthal angle of r. Finally we did not include the vector potential a since for an isolated vortex in an infinite area and in the London limit the effect of a is negligible as far as the low energy bound states are concerned. In the normal state, the hamiltonian leads to the band dispersion Ek = ±k − µ, where the sign depends on the chirality . Because of the spin-momentum locking the two bands are spin non-degenerate. Therefore the intra-band pairing must be odd and turns out to be p + ip′-like in the present case. This implies immediately a MF in the vortex core, just as in a p + ip′-superconductor18. The inter-band pairing is irrelevant for .

For our purpose, we need the wavefunctions of the low lying states. Following Volovik18, this is most easily obtained in the quasi-classical approximation. By rotation symmetry the states can by classified by the angular momentum l. The energy up to the first order of l is E = −lω0, with

where kF is the Fermi momentum and f0 is an envelop function

with a normalization constant C and s = r cos ϕ. The corresponding wavefunction is

where Jl is the Bessel function. The energy levels obtained here are essentially similar to the Caroli-de Gennes-Matricon states in usual s-wave superconductors19, except that the angular momentum l here is an integer rather than a half integer due to the fact that the helical state picks up a negative sign upon a rotation by 2π.

Of particular interest is the state with l = 0. The general argument presented in the previous section implies this zero energy state must describe a MF mode. Indeed, the creation field operator associated with this state is, up to the envelop function f0(r),

which is self-hermitian up to a trivial global phase. In principle there should be another MF mode on the outer boundary of an open system. But this is an artifact of the 2D model in view of the 3D heterostructure (where MF's appear on the top and bottom layers).

The electronic LDOS from the above states is given by

Since Jl(0) = 0 for l ≠ 0, the contribution to the spectral weight at r = 0 comes from l = 0 and l = 1 at energy ω = 0 and ω = ω0, respectively. Thus there are two LDOS peaks located asymmetrically in energy. The energy of the side peak is positive for µ > 0 here, but would be negative if µ < 0 instead. The asymmetry in the LDOS is a direct consequence of the absence of the particle-hole symmetry in the normal state band structure since the Fermi level is away from the Dirac point in our case. Similar asymmetries were actually found even for the bound states in the vortex core of usual s-wave superconductors as the Fermi level is close to the band edges, both theoretically and experimentally20,21. Of course, as the Fermi level approaches the Dirac point (µ → 0), kF decreases and thus the level spacing ω0 in our case increases. As kF → 0, the quasi-classical approximation fails but the tendency of ω0 ∝ 1/kF does imply the absence of in-gap bound states other than the unique Majorana zero mode for kF = 0, as discussed elsewhere15,16,17. But this is clearly not the case in the experiment, where many bound states are observed14.

As we step away from the vortex core, the other states with higher l contributes and from the behavior of Bessel functions, LDOS peaks should appear at the guiding centers of Jl,l±1(kFr) for ω = ±lω0. Such a peak splitting behavior is similar to that in usual superconductors. Rather unexpectedly the experiment reports a symmetric zero-bias peak and it appears that the central peak does not split until r is a few atoms away from the vortex core14. One possible reason for the discrepancy is the working temperature is higher than ω0 that smears out the asymmetry due to the side peak. Another possibility is impurities pinning the vortex may have suppressed the l = 1 state locally, leaving the l = 0 Majorana state intact. We shall come back to this point later.

Lattice model: We now consider an effective 2D lattice model for the low-lying states on the top layer of the heterostructure. The effective hamiltonian is,

where . We set tb = t = 1 and µ = 0.23. Notice that the σz terms are necessary to make sure that there is only one Dirac point in the Brillouine zone. Such terms violate time-reversal symmetry. Nonetheless, this effect is negligible for low lying states close to the Dirac point. In the presence of vortex, we couple the vector potential to electrons by minimal coupling as described in the previous section. We use a large lattice (of linear dimension L = 400 in this subsection) to reduce finite size effect. The pairing hamiltonian Hp is of a form closely similar to that in Eq.8, except that r is a 2D vector here. From now on we switch back to the convention in the Nambu space. The LDOS is given by

where

is the level-resolved local spectral weight. In practice, the delta function is smeared by a proper factor. On the other hand, the dimension of the single particle hamiltonian is too large to get all eigenstates. Instead, we use the real-space recursion method22,23 to get ρ(r, ω) most reliably.

Fig. 4(a) shows ρ(r, ω) for r along a (1, 1)-cut emanating from the vortex core and (b) shows a few low energy levels with the height proportional to An(r). There is clearly a ZBP and a side peak near the vortex core. As discussed previously, the zero energy state must be a MF mode. The energy of the side peak is E1 ~ Δ0/kFξ where kFξ ~ 2.3 in our case. Moreover, the peaks shift to higher energies as r is more distant from the core. All of these features are consistent with the previous analytical discussions. In particular, there should be a side peak accompanying the ZBP. It is however not observed in the experiment. Apart from thermal smearing, the discrepancy is likely caused by impurities which we consider in the following.

Figure 4
figure 4

(a) LDOS along a (1, 1)-cut, starting from the vortex core (bottom line). (b) Low energy levels with the height proportional to the respective spectral weight for each point along the cut.

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The impurity hamiltonian can be written as,

where V0(r) and Vi= 1,2,3(r) are the position dependent impurity strengths of scalar and magnetic impurities (of various polarizations), respectively. Here the single-particle part is denoted as Vimp(r) for convenience. We observe that {Vimp, τ1K} = 0. For a MF state |Ψ〉 (satisfying τ1K|Ψ〉 = |Ψ〉), the energy correction is, to the first order of Vimp,

which must be zero by comparing the first and last lines. Therefore the MF state is immune to weak impurities of any kind. In this sense the MF is both charge and spin neutral. On the other hand, no protection mechanism applies for the energy of other finite-energy bound states against Vimp. Fig. 5 is a similar plot to Fig. 4 but with a local scalar impurity right located at the vortex core. The impurity strength is V0 = 15Δ0. (This is a weak impurity since Δ0 is a small energy scale as compared to the bandwidth.) We observe that the previous side peak at energy E1 is completely wiped out. The higher energy levels remain intact since their wavefunctions vanish at the vortex core, thus are decoupled to the local impurity to the leading order. The LDOS becomes much more symmetric at the vortex core and the peak does not break near the vortex core, in nice agreement with the experiment14. We should emphasize, however, that the LDOS in our case is still slightly asymmetric near the gap edges, as shown in Fig. 5b. The asymmetry is an intrinsic feature of the model as the Fermi level is away from the Dirac point, as we discussed analytically and is enhanced by the relatively large gap scale in the calculation that is necessary to improve the numerical resolution. In the experiment, the gap scale is much smaller than the Fermi energy (the distance from the Fermi level to the Dirac point), so that the energy level spacing of the bound states is smaller. Combined with the finite temperature smearing, the experimental LDOS may appear more symmetric than what we show in Fig. 5. Therefore the difference to the theory is quantitative rather than qualitative.

Figure 5
figure 5

The same plot as Fig. 4 except that a local scalar impurity with V0 = 15Δ0 is located at the vortex core.

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We also show the corresponding results for a local magnetic impurity in Fig. 6. The impurity reduces the amplitude of the MF mode at the core, but the zero energy level remains robust. On the other hand, additional in-gap states are induced by the impurity. The LDOS is asymmetric in energy due to the side peaks. Experimentally neither the reduced ZBP nor the asymmetry of LDOS is observed, implying the absence of magnetic impurities therein.

Figure 6
figure 6

The same plot as Fig. 4 except that a local magnetic impurity with strength Vx = 12Δ0 is located at the vortex core.

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Finally we examine the effect of a finite concentration of impurities, an issue also relevant to experiments. Since the impurities too afar from the vortex core are irrelevant for the low-lying bound states, we put impurities within a radius R = 10 around the core randomly, with ensemble averages

where Wi is the strength of type i. For magnetic impurities we define as a measure of the total strength. By monitoring the presence/absence of the zero energy level, we obtain a phase diagram in Fig. 7, showing the critical strength Wc of W0 (Wm) for scalar (magnetic) impurities, above which the MF disappears, as a function of the impurity concentration c. The critical strength is reduced to the order of gap scale for both scalar and magnetic impurities. This is in contrast to the stability in the single impurity case. The result suggests that high sample quality is necessary.

Figure 7
figure 7

Wc versus c for scalar (red) and magnetic (blue) impurities.

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Summary

In conclusion, we provided strong theoretical support to the experimental result that there exists an optimal number (nopt) of TI layers for MF to appear on the top layer of the TI/SC heterostructure. We point out however that the MF modes on the top and bottom TI layers are bound to the vortex line and would cause difficulty to achieve braiding individual MF's. We show analytically and numerically that the LDOS near the vortex core has a ZBP and a side peak. The absence of the side peak in the experiment is nicely reconciled by the effect of a local scalar impurity pinning the vortex core. We also examined the effect of a finite concentration of impurities. While the MF mode is extremely robust to a single local impurity, the critical impurity strength drops drastically to the order of the effective pairing gap at a moderate impurity concentration. This points toward the importance of sample quality. We suggest that the effect of single and many impurities be tested by STM (at sufficiently low temperatures) intentionally to corroborate the MF mode further and to examine its stability, paving the way toward the application of such fascinating objects.