Existence of Majorana bound states near impurities in the case of a small superconducting gap

Physica E 89 (2017) 130–133

Contents lists available at ScienceDirect

Physica E journal homepage: www.elsevier.com/locate/physe

Existence of Majorana bound states near impurities in the case of a small superconducting gap

MARK

Yu.P. Chuburin Physical-Technical Institute, Ural Branch of Russian Academy of Sciences, Izhevsk, Russia

A R T I C L E I N F O

A BS T RAC T

Keywords: Topological insulator Zeeman field Majorana bound states Impurity Superconducting gap Transmission probability

We consider the edge states of a 2D topological insulator in the presence of the Zeeman field and in proximity to a s-wave superconductor. We analytically show that two linearly independent Majorana bound states (MBSs) can appear near the impurity located in a small region in which both the pairing parameter Δ and the Zeeman field M may be changed. We find two conditions for the existence of the MBSs: firstly, |Δ| ≈ |M |, ie, the superconducting gap in the spectrum should be sufficiently small; secondly, the absolute value of the average w of the impurity potential should have a certain value; the last condition is necessary. The equation |Δ| = |M | determines the boundary of the topological phase of the system, thus the system as a whole must be close to this boundary in relation to the parameters. If the same is true for the impurity region, then the second condition has the form w ≈ ± v /2 where v is the edge states velocity. In this case, the electron transmission probability is equal to 1 for energies close to zero.

1. Introduction In condensed matter, the Majorana bound states (MBSs) can be thought of as the zero-energy many-electron excitations; they are quasiparticles with no distinction between particles and antiparticles [1,2]. The MBSs may arise near domain walls between different topological phases in the so-called topological superconductors, in particular, at the ends of the quantum wire, proximity coupled to a superconductor [1–7]. This states are topologically protected, hence they are robust against extrinsic perturbations [2,5–7]. The MBSs may have good prospects of applications in quantum computing [1,5,6]. In theory, their existence is not in doubt, but their experimental observation is still being questioned [1,6]. In this paper we deal with the edge states of a 2D topological insulator in the presence of the Zeeman field and in proximity to a swave superconductor [1,2,5–7]. We will explore the possibility of the existence of MBSs near the impurity located in a small region in which both the pairing parameter Δ and the Zeeman field M may change. (See in [8,9] the general discussion on the existence of impurity-induced bound states near zero energy in 1D structures, proximity coupled to a superconductor). Using the Green function of the mean field Bogoliubov-de Gennes Hamiltonian (which we find explicitly), we analytically have proved that two linearly independent MBSs localized at the impurity can arise, but only if |Δ| ≈ |M |, ie, for a small superconducting gap in the spectrum, and, in addition, the absolute value of the average w of the impurity potential must have a certain value (see

below; the latter condition is necessary). We also obtain the explicit expressions for the wave functions of the MBSs. The equation |Δ| = |M | determines the boundary of the topological phase of our system [4], thus the system as a whole must be close to this boundary in relation to the parameters. If the same is also true for the impurity region, then w should be close to ± v /2 where v is the edge states velocity. In this case the electron transmission probability is equal to 1 for energies close to zero, but outside the gap (cf. the existence of the zero energy conductance peak [3]). Another possibility is discussed in Section 3. The results may be useful for the experimental observation of the MBSs. 2. Spectrum and green function In this section, we study the following Bogoliubov-de Gennes Hamiltonian [1,2,4,10]:

⎛− ivσx ∂x + Mσz ⎞ Δiσy H=⎜ ⎟ − Δiσy ivσx ∂x − Mσz ⎠ ⎝ where σx , σy , and σz are the Pauli matrices acting in the spin space, the pairing amplitude Δ ≠ 0 is assumed to be real, and M = const . Further, we set v=1. The wave functions of the Hamiltonian H have the form

ψ = (ψ1↑, ψ1↓, ψ2↑, ψ2↓) = (ψ1, ψ ′1 , ψ2, ψ ′2 ). Here the components with index 1 and 2 refer to particles and holes,

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.physe.2017.02.017 Received 17 October 2016; Received in revised form 18 January 2017; Accepted 20 February 2017 Available online 21 February 2017 1386-9477/ © 2017 Elsevier B.V. All rights reserved.

Physica E 89 (2017) 130–133

Y.P. Chuburin

respectively. To investigate the Majorana states and their influence on the scattering pattern, we will use the Green function of the Hamiltonian H. We will find this function by solving the equation (H − E ) ψ = φ or, in more detail,

−i ∂x ψ ′1 − (E − M ) ψ1 + Δψ ′2 = φ1,

−

∫−∞

∞

∫−∞ (eip |x−x′|/p1 − eip |x−x′|/p2 ) φ′2 (x′) dx′, 1

2

∫

∫

with respect to ψ. After Fourier transformation

l ( p) = ψ (x ) ↦ ψ

E+M−Δ 4i

2

∫

i ∂x ψ2 − (E − M ) ψ ′2 + Δψ1 = φ′2 ,

e−ipx ψ

+

1

∫

(1)

i ∂x ψ ′2 − (E + M ) ψ2 − Δψ ′1 = φ2 ,

∞

∫−∞ (eip |x−x′| − eip |x−x′|)sgn(x − x′) φ2 (x′) dx′

ψ ′1 (x ) = ((H − E )−1φ)′1 (x ) ∞ E−M =− eip1| x − x ′|φ′1 (x′) dx′ −∞ 2ip1 ∞ 1 − eip1| x − x ′| sgn(x − x′) φ1 (x′) dx′ 2i −∞ ∞ Δ − eip1| x − x ′|φ2 (x′) dx′ 2ip1 −∞ E − M − Δ ∞ ip1| x − x ′| + (e / p1 −∞ 4i

−i ∂x ψ1 − (E + M ) ψ ′1 − Δψ2 = φ′1 ,

1 2π

∞

1 4i

ip2 | x − x ′|

−e 1 + 4i

(x ) dx

+

1

2

(7)

∞

∫

∫

another two equations are obtained from (6), (7) by replacing (2)

ψ1 → −ψ ′2 , ψ ′1 → ψ2, φ1 → −φ′2 , φ′1 → φ2 , φ2 → φ′1, φ′2 → −φ1.

We denote by d = d ( p ) the determinant of the system (2). Then

For E not belonging to the spectrum, the signs in (5) are determined by the decrease of the exponential functions. If E belongs to the spectrum, these signs determine the direction of movement of the particles.

d = E 4 − 2E2 (M2 + p 2 + Δ2 ) + (M2 + p 2 − Δ2 )2 = p 4 − 2p 2 (E2 − M2 + Δ2 ) + (E2 − M2 − Δ2 )2 − 4M2Δ2 .

∫−∞ (eip |x−x′| − eip |x−x′|)sgn(x − x′) φ1 (x′) dx′

1 (eip1| x − x ′| − eip2 | x − x ′|)sgn(x − x′) φ′2 (x′) dx′ 4i −∞ −E + M + Δ ∞ ip1| x − x ′| + (e / p1 − eip2 | x − x ′| / p2 ) φ2 (x′) dx′; −∞ 4i

we obtain from (1)

⎛ − (E − M ) ⎞⎛ ψ l1 ⎞ ⎛ φ l1 ⎞ p 0 Δ ⎜ ⎟⎜ ⎟ ⎜ ⎟ l l p − ( E + M ) − Δ 0 ψ ′ φ ′1 ⎟ 1 ⎜ ⎟ ⎜ ⎜ ⎟ = . 0 −Δ − (E + M ) −p ⎜ ⎟⎜ ψ l2 ⎟ ⎜ φ l2 ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ Δ 0 −p − (E − M ) ⎠ ⎝ ψ ⎝ l′2 ⎠ ⎝ φ l′2 ⎠

/ p2 ) φ′1 (x′) dx′

∞

(3)

By (3), the equation d=0 is equivalent to the equations E2 = (Δ ± M2 + p 2 )2 . Hence the spectrum of H is the union of (−∞, −α ] and [α , ∞) where α = min{|M − Δ|, |M + Δ|}. Also from (3) we have

3. Results and discussion

⎛ ⎞ 1 1 1 1 1 ⎜ ⎟, = 2 = 2 − 2 2 2 2⎜ 2 2 2⎟ 2 d ( p − p1 )( p − p2 ) p1 − p2 ⎝ p − p1 p − p2 ⎠

According to [8], the nonmagnetic impurities may lead to subgap bound states in wires only in the presence of a combination of Zeeman splitting and Rashba spin-orbit coupling, which are also required to realize a topological superconducting phase. In our case, the role of the Rashba interaction plays the 1D Dirac Hamiltonian. (Note that it can be directly checked that for −d 2 / dx 2 instead of the Dirac Hamiltonian, the MBSs do not arise near the impurity). Let us write the equation describing the eigenfunctions of the Hamiltonian H + V where V is the potential, corresponding to the energy E,

3.1. Majorana states

(4)

p12 − p22 = 4EΔ where

p1 = ± (E + Δ)2 − M2 , p2 = ± (E − Δ)2 − M2 .

(5)

First, performing the calculations like [11], we find from (2) and Cramer's rule the Green function of H in the momentum representation and then, using (4) and the known formulas

1 2π 1 2π

∫−∞ e p2φl−( pa) 2dp ∞

∞

∫−∞ e

ipx

=−

ipx pφ l ( p ) dp

p 2 − a2

⎛ λM + V0 0 0 − νΔ ⎞ ⎜ ⎟ 0 − λM + V νΔ 0 0 ⎟ δ (x ) V=⎜ 0 νΔ − λM − V0 0 ⎜ ⎟ ⎜ ⎟ 0 0 λM − V0 ⎠ ⎝ − νΔ

∫−∞ eia|x−x′|φ (x′) dx′, ∞

1 2i

∫−∞ eia|x−x′| sgn(x − x′) φ (x′) dx′,

(9)

where λ , ν , and V0 are arbitrary real constants, modeling the change of the Zeeman field and the pairing parameter and also the presence of the impurity near x = 0. We note that w = 4V0 is the average of the impurity potential (for λ = ν = 0 ). In the results obtained below, we can replace δ (x ) with smooth non-negative even function with a support in a sufficiently small neighborhood of zero, the integral of which is equal to 1. Next, we consider only the even smooth approximation of the Dirac function δ (x ), which corresponds to the symmetrical distribution of values of V around zero. To find the MBSs, we set E=0 in (8), and therefore in the expressions for the Green function (see (6), (7) and the remark after (7)). Then, by (5), p1 = p2 = p = ± Δ2 − M2 . To obtain the decrease of the eigenfunctions at infinity, we have to assume that |M | > |Δ| (see [8] where the authors show the necessity of this condition for the existence

we obtain it in the coordinate representation. As a result, we get the following expressions: ∞ E+M eip1| x − x ′|φ1 (x′) dx′ −∞ 2ip1 ∞ 1 eip1| x − x ′| sgn(x − x′) φ′1 (x′) dx′ − 2i −∞ ∞ Δ eip1| x − x ′|φ′2 (x′) dx′ + 2ip1 −∞ E + M − Δ ∞ ip1| x − x ′| + (e / p1 − eip2 | x − x ′| / p2 ) φ1 (x′) dx′ −∞ 4i ∞ 1 + (eip1| x − x ′| − eip2 | x − x ′|)sgn(x − x′) φ′1 (x′) dx′ 4i −∞

∫

ψ1 (x ) = ((H − E )−1φ )1 (x ) = −

(8)

We use the short-range potential of the form

∞

1 2ia

=−

ψ = −(H − E )−1Vψ .

∫

∫

∫

∫

(6) 131

Physica E 89 (2017) 130–133

Y.P. Chuburin

V02 = 2 2 |Δ|3/2 (λ + ν )(|M | − |Δ|)−1/2 + O (1) ∼ (|M | − |Δ|)−1/2 → ∞

of subgap impurity bound states) and p = i M2 − Δ2 . For |M | < |Δ| this eigenfunctions increase exponentially (see (10) below); this means their short lifetime, such eigenfunctions describe the resonance states. Eq. (8) takes the form M2 − Δ2 | x|

e−

ψ1 (x ) =

2 M2 − Δ2

ψ ′1 (x ) =

e−

as |M | − |Δ| → 0 . In the case λ = −ν , for |M | ≈ |Δ| and V02 ≈ 4 , from (14) we obtain the wave functions

[−(λM2 + MV0 + νΔ2 ) ψ1 (0) + Δ ((λ + ν ) M − V0 ) ψ ′2 (0)

− i M2 − Δ2 sgn(x )((− λM + V0 ) ψ ′1 (0) + νΔψ2 (0))]; M2 − Δ2 | x|

2 M2 − Δ2

ψ ′1 (x ) ≈ C (−1, i sgn(x ), −i sgn(x ), 1)T e− (10)

ψ ′2 (x ) ≈ C (sgn(x ), i , −i , −sgn(x ))T e−

ψ2 (x ) =

e

M2

2

−

Δ2

e−

M2 − Δ2 | x|

2 M2 − Δ2

ψ ′2 (x ) ≈ C (sgn(x ), 0, 0, −sgn(x ))T e−

[Δ ((λ + ν ) M − V0 ) ψ ′1 (0) − (λM2 + MV0 + νΔ2 ) ψ2 (0)

[Δ ((λ + ν ) M + V0 ) ψ1 (0) − (λM2 − MV0 + νΔ2 ) ψ2′

⎛ c+ ⎜ ⎜0 ⎜0 ⎜ ⎝ d+

0 c− d− 0

0 d+ c+ 0

d− ⎞ ⎛ ψ1 (0) ⎞ ⎟ ⎟⎜ 0 ⎟ ⎜ ψ ′1 (0) ⎟ =0 0 ⎟ ⎜ ψ2 (0) ⎟ ⎟⎜ ⎟ c− ⎠ ⎝ ψ ′2 (0)⎠

M2

Δ2

λM2

(11)

νΔ2

− + + ± MV0 , d ± = −Δ ((λ + ν ) M ± V0 ). where c ± = 2 The determinant of this system (11) is D = [(2 M2 − Δ2 + λM2 + νΔ2 )2 − (MV0 )2 − Δ2 ((λ + ν )2 M2 − V02 )]2 . The determinant D is equal to zero, and therefore, there exist non-zero solutions of (11), if

V02 =

(2 M2 − Δ2 + λM2 + νΔ2 )2 − Δ2 M2 (λ + ν )2 . M2 − Δ2

(12)

If (12) holds, there are two linearly independent solutions of the system (11) of the form

(2 M2 − Δ2 + λM2 + νΔ2 − MV0 , 0, 0, Δ ((λ + ν ) M + V0 ))T ,

M2 − Δ2 | x|.

(18)

(19)

[1], hence, they are the MBSs; furthermore, they are orthogonal. We note that instead of the “blurred” zero energy of overlapping Majorana states [3], we have here the “blurred” condition above, if |M | − |Δ| is not small enough. Thus if (12) holds, then the bounded zero-energy states emerge in the neighborhood of the impurity, which transform into the MBSs with decreasing the gap length. (According to (12), (19) we have 5 real equations and 5 parameters V0, M , Δ, λ , and ν to determine the MBSs. It is understood that the exact solutions may not exist). The MBSs corresponding the equation λ = −ν are more stable with respect to the width of the gap, by (15) compared with (16). This equation means that in the neighborhood of x=0 for the new values M ′ = (1 + λ ) M , Δ′ = (1 − ν ) Δ we have |M ′| ≈ |Δ′|. The equation |Δ| = |M | determines the boundary of the topological phase for the system as a whole [4], and according to our assumption, the system is close to this boundary in relation to the parameters. Hence if this persists for the impurity region, then the MBSs are stable in the sense above. Therefore, here the concept of the topological phase retains its importance for the small regions. Note that the degree of localization of the MBSs decreases as |M | → |Δ|, and the MBSs become delocalized at |M | = |Δ|. (This is not true in the case of normal conductor-superconductor hybrid systems [12]). To obtain the spatially separated states, we can use two impurity regions.

ψj(′) (0),

ψj(′) (−0))/2 .

M2 − Δ2 | x|,

ψ2′* = −ψ1, ψ2* = ψ ′1

= + We put To find we multiply (10) by the smooth even approximating functions δm (x ) → δ (x ), m → ∞. Next, taking the integral and passing to the limit as m → ∞, we obtain the following system of equations with respect to ψj(′) (0): (ψj(′) (+0)

(17)

The wave functions (17), (18) satisfy (8) for E=0 and the condition for Majorana states

+ i M2 − Δ2 sgn(x )(−(λM + V0 ) ψ2 (0) + νΔψ ′1 (0))]. ψj(′) (0)

M2 − Δ2 | x|.

ψ ′1 (x ) ≈ C (0, i sgn(x ), − i sgn(x ), 0)T e−

+ i M2 − Δ2 sgn(x )((λM − V0 ) ψ ′2 (0) − νΔψ1 (0))]; ψ ′2 (x ) =

M2 − Δ2 | x|,

where C = const . Similarly, in the case λ ≠ −ν , for |M | ≈ |Δ| and V02 → ∞, we have

[(−λM2 + MV0 − νΔ2 ) ψ ′1 (0) + Δ ((λ + ν ) M + V0 )

ψ2 (0) − i M2 − Δ2 sgn(x )((λM + V0 ) ψ1 (0) − νΔψ ′2 (0))]; − M2 − Δ2 | x|

(16)

(13) 3.2. Scattering near MBSs

(0, 2 M2 − Δ2 + λM2 + νΔ2 + MV0, Δ ((λ + ν ) M − V0 ), 0)T . Let us study the scattering problem for the potential V, using the Lippmann-Schwinger equation ψ = ψ0 − (H − E )−1Vψ where (H − E ) ψ0 = 0 . We now assume that

For the corresponding wave functions, according to (10), we derive the expressions ⎛ ⎞ 2 2 2 2 2 2 2 2 2 ⎜ − (λM + νΔ + MV0 ) + M − Δ (− λ M + ν Δ + V0 )/2 ⎟ 2 ⎜ ⎟ 2 2 2 2 2 2 ψ1 (x ) = ⎜− i sgn(x )[(λM + V0 ) M − Δ + M (λ M − ν Δ − V0 )/2]⎟ e− ⎜ ⎟ i sgn(x )[− νΔ M2 − Δ2 + Δ (λ2M2 − ν 2Δ2 − V02 )/2] ⎜ ⎟ Δ ((λ + ν ) B + V0 ) ⎝ ⎠

Δ ≫ E ≫ M − Δ > 0, M2 − Δ2 | x|,

and E is so close to zero that we can use the Eq. (10) for Green function. Thus, |M | − |Δ| ≈ 0 , and the gap is very small. First we find ψ0 as the solution of (1), for φ = 0 and p = p1 = (E + Δ)2 − M2 . We have

(14) ⎞ ⎛ 2 2 2 2 2 2 2 ⎜− i sgn(x )[(− λM + V0 ) M − Δ + M (− λ M + ν Δ + V0 )/2]⎟ ⎜ − λM2 − νΔ2 + MV + M2 − Δ2 (− λ2M2 + ν 2Δ2 + V 2 )/2 ⎟ 0 0 ψ2 (x ) = ⎜ ⎟ e− Δ ((λ + ν ) B − V0 ) ⎟ ⎜ ⎟ ⎜ i sgn(x )[νΔ M2 − Δ2 + Δ (− λ2M2 + ν 2Δ2 + V02 )/2] ⎠ ⎝

⎛ E+Δ+M ⎞ ⎛ 2Δ⎞ ⎜ ⎟ 2 2 ⎜ ⎟ ⎜ (E + Δ) − M ⎟ ipx ψ0 (x ) = 2EΔ ⎜ e ≈ 2EΔ ⎜ 0 ⎟ eipx 2 − M2 ⎟ E Δ ( + ) ⎜0⎟ ⎜⎜ ⎟⎟ ⎝ 2Δ⎠ ⎝ − (E + Δ + M ) ⎠

M2 − Δ2 | x|.

or, after normalization, keeping the same notation,

It is easy to see that if λ = −ν , then

lim |M |−| Δ |→0

V02

=4

or, returning the edge state velocity v, spectral gap is small). If λ ≠ −ν , then

ψ0 (x ) = (1, 0, 0, −1)T eipx / 2 .

(15)

V02

≈

4v 2

as |M | ≈ |Δ| (ie, the

(20)

Using (10), (20), and the assumptions above, we can write the Lippmann-Schwinger equation: 132

Physica E 89 (2017) 130–133

Y.P. Chuburin

ψ1 (x ) =

eipx eip | x| − [−Δ ((λ + ν ) Δ + V0 ) ψ1 (0) + Δ ((λ + ν ) Δ − V0 ) ψ ′2 (0)], 2ip 2

impurities with different average potentials). Then there are two linearly independent MBSs near the impurity, which are stable with respect to the gap width. The explicit expressions for the wave functions of the MBSs were obtained. Under the above conditions, the transmission probability of electrons is close to 1 for small energies. To obtain the rigorous analytical formulas, we had to use a simple zero-range potential. This led to the localization of two linearly independent states in the same region. In the case of a more realistic finite-range potential, we suppose that there is a small Zeeman field λM ≈ M − Δ in the (sufficiently small) impurity region such that there is another topological phase in this region than in the whole system. (By (12), this does not affect the existence of the MBSs). Then we can expect that the MBSs will be spatially separated and, therefore, will be stable with respect to external fields. Indeed, in this case the boundary points of the impurity region separate the areas with different topological phases, and thus this points “attract” the MBSs [4]. To verify the existence of MBSs, we must ensure that the MBSs contribution to the conductance (see the end of Chapter 3) is not zero. According to (12), (24), the sufficiently large local Zeeman field λM destroys the MBSs and reduces the MBSs contribution to the conductance to a negligible value. At the same time, we can expect that this local field does not significantly affect the “non-topological” superconductivity [13,14]. Therefore, the decrease of the conductance with increasing |λ| indicates the existence of MBSs.

(21) eip | x| [− Δ ((λ + ν ) Δ − V0 ) ψ ′1 (0) + Δ ((λ + ν ) Δ + V0 ) ψ2 (0)], ψ ′1 (x ) = − 2ip ψ2 (x ) = − ψ ′2 (x ) = −

eip | x| [Δ ((λ + ν ) Δ − V0 ) ψ ′1 (0) − Δ ((λ + ν ) Δ + V0 ) ψ2 (0)], 2ip eipx eip | x| − [Δ ((λ + ν ) Δ + V0 ) ψ1 (0) − Δ ((λ + ν ) Δ − V0 ) ψ ′2 (0)]. 2ip 2

From this we find the linear system with respect to ψj(′) (0)

⎛ α+ ⎜ ⎜0 ⎜0 ⎜ ⎝ β+

0 α− β− 0

0 β+ α+ 0

β− ⎞ ⎛ ψ1 (0) ⎞ ⎟⎜ ⎟ 0 ⎟ ⎜ ψ ′1 (0) ⎟ = 0 ⎟ ⎜ ψ2 (0) ⎟ ⎟ ⎟⎜ α− ⎠ ⎝ ψ ′2 (0)⎠

⎛ 1 ⎞ ⎜ ⎟ 2 ip ⎜ 0 ⎟ ⎜ 0 ⎟ ⎝− 1⎠

(22)

where α ± = 2ip − Δ ((λ + ν ) Δ ± V0 ), β ± = Δ ((λ + ν ) Δ ± V0 ). We denote by aj(′)+ and aj(′)− the transmission and reflection amplitudes, respectively, for each component of the wave vector. By (21), we have

aj(′)+ = ψj(′) (0), j = 1, 2; a1− = ψ1 (0) −

1 1 ; a 2′− = ψ ′2 (0) + ; 2 2

a1′− = a1′+; a 2− = a 2+.

(23)

Using (22), (23) and Cramer's rule, we find

a1+

=

ip , a1− = 2 (ip − Δ2 (λ + ν ))

a 2′+ = −a1+ = −

Acknowledgments

Δ2 (λ + ν ) , 2 (ip − Δ2 (λ + ν ))

ip , a 2′− = a1− = 2 (ip − Δ2 (λ + ν ))

This work was partially supported by the Ural Branch of the Russian Academy of Science grant No. 15-8-2-12.

−Δ2 (λ

+ ν) , 2 (ip − Δ2 (λ + ν ))

References

a1′± = a 2± = 0.

[1] R. Elliott Steven, M. Franz, Colloquium: Majorana fermions in nuclear, particle, and solid-state physics, Rev. Mod. Phys. 87 (2015) 137. [2] T.D. Stanescu, S. Tewari, Majorana fermions in semiconductor nanowires: fundamentals, modeling, and experiment, J. Phys.: Condens. Matter 25 (2013) 233201. [3] C.-H. Lin, J.D. Sau, S.D. Sarma, Zero-bias conductance peak in majorana wires made of semiconductor/superconductor hybrid structures, Phys. Rev. B 86 (2012) 224511. [4] T. Karzig, G. Refael, F. von Oppen, Boosting majorana zero modes, Phys. Rev. X 3 (2013) 041017. [5] J. Alicea, New directions in the pursuit of majorana fermions in solid state systems, Rep. Prog. Phys. 75 (2012) 076501. [6] M. Sato, S. Fujimoto, Majorana fermions and topology in superconductors, J. Phys. Soc. Jpn. 85 (2016) 072001. [7] C.W.J. Beenakker, Search for majorana fermions in superconductors, Condens. Matter Phys. 4 (2013) 113. [8] J.D. Sau, E. Demler, Bound states at impurities as a probe of topological superconductivity in nanowires, Phys. Rev. B 88 (2013) 205402. [9] X.J. Liu, Impurity probe of topological superfluids in one-dimensional spin-orbitcoupled atomic fermi gases, Phys. Rev. A 87 (2013) 013622. [10] J. Linder, Y. Tanaka, T. Yokoyama, A. Sudbo, N. Nagaosa, Interplay between superconductivity and ferromagnetism on a topological insulator, Phys. Rev. B 81 (2010) 184525. [11] Yu.P. Chuburin, Electron scattering in a graphene nanoribbon in the presence of ferromagnetic layer and rashba interaction, Phys. Lett. A 380 (2016) 1329. [12] D. Sticlet, B. Nijholt, A. Akhmerov, arXiv preprint, Robustness of Majorana bound states in the short junction limit, arXiv:1609.00637, 2016, arxiv.org. [13] H. Shiba, A hartree-fock theory of transition-metal impurities in a superconductor, Prog. Theor. Phys. 50 (1973) 50. [14] A.V. Balatsky, I. Vekhter, J.-X. Zhu, Impurity-induced states in conventional and unconventional superconductors, Rev. Mod. Phys. 78 (2006) 373.

Thus only spin-up electrons may participate in the scattering process. The transmission probability is equal to

P + = |a1+|2 + |a 2′+|2 =

(E + Δ)2 − M2 2E ≈ . (E + Δ)2 − M2 + Δ4 (λ + ν )2 2E + Δ3 (λ + ν )2 (24)

P+

= 1, therefore, such MBSs contribute In the case λ = −ν we have to the conductance. This looks similar to the zero energy conductance peak [3], but for E outside the gap. If 2E ⪡Δ3 (λ + ν )2 , ie, “far from stability”, we have P + ≈ 0 . Hence for sufficient large λ = −ν or Δ, to stop the electron current, we may change ν by −ν . 4. Conclusion In this paper, we analytically show the existence of the MBSs near the single impurity, at the edge of the 2D topological insulator with the proximity induced pairing Δ and in the presence of the Zeeman field M; in the small neighborhood of the impurity these quantities may be different from those outside the neighborhood. Assume that 1) Δ ≈ M both in the neighborhood of the impurity and outside it (therefore the superconducting gap is small); 2) |w| ≈ |v|/2 where w is the average of the impurity potential and v is the edge state velocity (note that this condition can be satisfied if there is a sufficiently large number of

133