Linear giant Stark effect in WSe2 nanoribbons

Physica B: Condensed Matter 545 (2018) 159–166

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Linear giant Stark effect in WSe2 nanoribbons

T

Mohsen Shahbazi Department of Physics, Factualy of Science, University of Zanjan, Zanjan 45371-38791, Iran

A R T I C LE I N FO

A B S T R A C T

Keywords: WSe2 nanoribbon Tight binding Electric field Stark effect Spin polarization Exchange interaction

In this paper the electronic properties of WSe2 nanoribbons is theoretically and numerically investigated by using six-band tight-binding (TB) model. Zigzag and armchair models of WSe2 nanoribbon show semiconducting and metallic phases, respectively. Effect of external electric field on the electronic properties of WSe2 via solving the Laplace's equation is also studied. A semiconductor-metal phase transition is observed for armchair structure V by 2.1 w of applied transverse electric field, where w is the width of the ribbon. A linear giant Stark effect is also occurred in both armchair and zigzag WSe2 nanoribbons by variations of applied transverse electric field. Moreover, we investigate electronic transport of armchair WSe2 nanoribbon putting on a ferromagnetic EuS substrate using nonequilibrium Green's function method. Spin polarization is occurred by effect of an exchange field of 0.2 eV, induced by a ferromagnetic EuS substrate.

1. Introduction

been utilized to investigate electronic and transport properties of WSe2 nanoribbons. Slater-Koster approach have also been employed to calculate hopping energies in a monolayer of WSe2 [13]. Notice that due to the presence of horizontal mirror symmetry for monolayer of WSe2, a reduced six-band Hamiltonian is utilized in this work. Furthermore, we evaluate the variations of electronic properties of zigzag and armchair WSe2 nanoribbons in the presence of transverse electric field. Linear giant Stark effect is also occurred by varying the external electric field in both zigzag and armchair WSe2 nanoribbons. Moreover, effect of an induced exchange field via a magnetic EuS substrate is perused. This paper is organized as follow. In Sec. 2, tight binding model that is utilized to calculate the energy band structure for both nanosheet and nanoribbon is explained. In Sec. 3, the Green's function approach is described to calculate the conductance and spin polarization. In Sec. 4, we investigate effect of external electric field and also an induced exchange field on the initial tight binding Hamiltonian of WSe2 singlelayer. In Sec. 5, we present our main results, including energy dispersion in the systems, energy dispersion in presence of external electric field, variations of energy band gap as a function of external electric fields, and eventually conductance and spin polarization in the presence of an induced exchange field by ferromagnetic EuS substrate. Finally, we summarize our main results in Sec. 6.

Two dimensional (2D) materials similar to graphene and monolayer of the transition metal dichalcogenides (TMDs) that recently have been introduced, are in the attention of scientists due to their exciting electronic properties. These materials have high carrier mobility [1] and have very high flexibility under strain so that their electronic properties could be tuned by strain engineering [2,3]. Notice that the intrinsic energy band gap of TMDs and graphene is different [4]. Moreover, TMDs, similar to MoS2, could possess lattice defects that can be helpful to tune their electronic properties by defect management [5]. Generally, TMDs are marked with MX2, where M stands for Mo or W, and also X stands for Te, S and Se. Recently electronic devices such as Field Effect Transistors (FET), sensors [6], and optical transistors [7] have been introduced based on TMDs. By thinned down from bulk to monolayer in TMDs, the inverse symmetry is broken [8]. There is an indirect energy gap of 0.94 eV for bulk WSe2, while a direct band gap of 1.61 eV in its monolayer is observed by Kumar et al. [4]. In WSe2 monolayer the Tungsten layer stands between two layers of Selenium via covalent bonds [9]. By means of external perturbations, it is possible to control the electronic and optical properties of a monolayers of WSe2. For example, it is possible to control the valley degree of freedoms by applying external magnetic fields [10]. Valley Zeeman effect has experimentally been observed by applying external magnetic field in WSe2 monolayer [11]. The giant Zeeman-type spin polarization can also be tuned by exerting external electric field in WSe2 monolayer [12]. In this paper, both tight binding and Green's function methods have

E-mail address: [email protected]. https://doi.org/10.1016/j.physb.2018.06.004 Received 9 January 2018; Received in revised form 22 May 2018; Accepted 3 June 2018 0921-4526/ © 2018 Elsevier B.V. All rights reserved.

2. Theory and methodology To determine electronic state of crystalline materials different techniques can be used. Tight binding, using the well-known Bloch theory, is a powerful technique for this purpose. That is related to the

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Fig. 1. (a) Top view and (b) side view of a WSe2 monolayer. Green and red circles indicate W and Se atoms respectively. u i and vi refer to nearest and next nearest neighbors, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

where, ci†. μ create an electron in the unit cell i in the atomic orbital μ. Also ∈ and t denote onsite and hopping energies respectively. Using Fourier transformation, the six-band Hamiltonian in real space (Eq. (4)), can be transformed to the k space as following:

Table 1 Tight binding parameters of WSe2 monolayer. All energies are in eV unite. Crystal Fields

f2 fp

− 0.935 − 2.321 − 5.629

fz

− 6.759

W− Se

Vpdσ Vpdπ

5.803 − 1.081

W− W

Vddσ Vddπ Vddδ Vppσ Vppπ

− 1.129 0.094 0.317 1.530

f0

Se − Se

3

H TB (k ) = ε +

− 0.123

)

P Sx =

1 (pt + pbx ), 2 x

P Sy =

1 ⎛ t b⎞ ⎜p + py ⎟, 2⎝ y ⎠

(2)

P zA =

1 (pt − pzb), 2 z

(3)

(1)

∑ ∈μν i . μν

ci†. μ ci . ν +

∑ ij . μν

t iXMeik·ui

H= H00 + eik aH−10 + e−ik aH−† 10,

t iMX e−ik·ui

⎤, 2t iXXcos(k ·vi ) ⎦

(5)

(6)

where H00 is Hamiltonian of a unit cell indicated in Fig. 1(a), and H−10 is the Hamiltonian between H00 and its adjacent unit cell. Notice that we consider n = 36 and n = 38 atoms for armchair and zigzag WSe2 nanoribbons unite cell, respectively. So, the ribbon width for armchair and zigzag WSe2 nanoribbons are 29.34 A0 and 49.27 A0, respectively.

where t and b imply to the top and bottom P orbitals respectively. pt and pb refer to symmetric and antisymmetric of p orbitals respectively. The even subspace of d orbitals of the Tungsten atom is also (dxy , d x2− y 2 , d3z2 − r 2). Eventually, the Hilbert space in even subspace for WSe2 monolayer is (dxy , d x2− y 2 , d3z2 − r 2 , P Sx , P Sy , P zA). (more details is in ref [8]). Based on this descriptions, the six-band Hamiltonian in real space can be explained as:

H TB =

2t iMMcos(k ·vi )

where ε = diag(εM, εX ) are onsite energy matrices, u and v denote nearest and next nearest neighbors respectively, (see Fig. 1). Moreover M and X refer to W and Se atoms, respectively. It should be mentioned that distance between nearest neighbor in plane W-W and Se-Se is 3.26 A0 and the out of plane bound length of W-Se is 2.21 A0 [14]. Tight binding parameter including crystal fields fα and hopping terms Vα of the WSe2 single-layer are calculated by fitting the low energy of conduction and valence bands [8], and the results are provided in Table 1. It should be noted that the crystal fields f0 , f1 , f2 refer to the atomic levels of the Tungsten orbitals consist of l = 0 (d3z2 − r 2), l = 1(dxz, dyz) and l = 2 (dxy ,d x2− y 2) respectively. Due to dxz , dyz orbitals are not involved in six-band model, the f1 is not included in Table 1. Also fp and fz denote the atomic level of the in-plane px , py and out-of-plane pz of the Selenium atom respectively. By using these parameters, the hopping and onsite energies of WSe2 single-layer for six-band tight binding model have been calculated using and results are provided in Appendix A. To obtain the Hamiltonian in the k space for WSe2 nanoribbon, the Fourier transformation of Eq. (4) must be only in one direction. Therefore, the Hamiltonian of WSe2 nanoribbons in k space can be written as following:

linear combination of atomic orbitals (LCAO). Single-layer of WSe2 has honeycomb lattice structure in top view (Fig. 1(a)) and three layers in side view, Fig. 1(b). A unit cell is considered including one Tungsten atom in the middle and two Selenium atoms in top and bottom, see Fig. 1(b). Hence, five d orbitals of Tungsten and three p orbitals of each of the Selenium atoms form the Hilbert space. Therefore, this Hilbert space can be read as dxy , dyz, dxz, d x2− y 2 , d3z2 − r 2 , ptx , pty , ptz , pbx , pby , pzb which build eleven-band tight binding model. A unitary transformation can be utilized due to the horizontal reflection symmetry proportional to z-axis so that this 11 × 11 matrix of Hamiltonian can be divided to even and odd subspace. The even subspace of the P orbitals can be written as following:

(

∑⎡ i=1 ⎣

3. Green's function method Quantum transport parameters of nanoribbons provide us information about electric current along the ribbon direction and also the number of channels of the electric current. The system that is considered to investigate the electronic transport is a WSe2 nanoribbon. In order to calculate the conductance of WSe2 nanoribbons, we use the non-equilibrium Green's function method (NEGF) which have been extended by Lopez et al. [15]. In this method, ribbon is divided to three

[t ij . μν ci†. μ cj . ν + H.C] , (4) 160

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Fig. 2. Zigzag WSe2 nanoribbon as a quantum transport device. Green and red circles indicate W and Se atoms respectively. Notice that there are two different edges named W and Se edges in zigzag WSe2 nanoribbon. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

4. Effects of electric field and an exchange interaction

parts including left-electrode, device and right-electrode (Fig. 2.). So the total Hamiltonian of this system can be written as following:

In order to investigate electronic and transport properties of WSe2 monolayer in the presence of external transverse electric field, we solve the Laplace's equation in the ribbon space. Thus, the tight binding Hamiltonian in the presence of transverse electric field can be written as following:

(7)

H= HD − ΣL − ΣR ,

where ΣL and ΣR denote left and right self-energies respectively. Actually, the self-energies originate only from connections between the device and electrodes. It should be noted that device and electrodes are considered as several unit cells (see Fig. 2). The self-energies are calculated from following equations:

H TB =

∑ ∈μν

c†i·μ ci·ν +

i·μν

L ΣL = H†01g00 H01,

(8)

ΣR = Hm.m+ 1gmR+1.m+ 1 H†m.m+ 1,

(9)

(10)

where s refer to spin of electron and η denote so small number close zero used to eliminate singularity. The transport parameters including conductance and transmission function for each spin component can be calculated using following equations:

gs =

e2 s T, h

T s = Tr(ΓL G s ΓR G s †),

Σ†L)

H TB =

g↑ − g↓ , g↑ + g↓

DOS =

g↑

where and spectively.

g↓

(15)

c†i·μ ci·ν +

∑ [tij·μνc†i·μ cj·ν + H.C] + λ ∑ c†i·μ σz ci·μ, ij·μν

i, μ

(16)

(11)

where λ and σz denote exchange field and z-component of Pauli matrices respectively [19].

(12)

5. Numerical results and discussion

Σ†R )

The Hamiltonian of Eq. (5) leads to the six bands in energy dispersion of WSe2 nanosheet. After diagonalizing Eq. (5), band structure for WSe2 nanosheet is calculated and results are shown in Fig. 3. As it is clear, six-band model in the energy dispersion curve ofWSe2 is noticeable. Notice that in this paper Fermi energy is set to Zero. A direct band gap of 1.74 eV is occurred in K point which is comparable with the previous result obtained by Kumar et al. [4]. The energy band structure of WSe2 nanoribbons (and other transition metal dichalcogenides) is different from the energy dispersion of WSe2 nanosheet due to existence of edge modes [20]. We calculate energy band structure for both zigzag and armchairWSe2 nanoribbons by

(13)

−1 Im(Tr(G )) π

∑ ∈μν i·μν

and ΓR = i(ΣR − are the broadening matrices where ΓL = i(ΣL − with the left and right handed terminal respectively. Spin polarization (ρ) and density of states (DOS) can also be calculated as following equations:

ρs =

i, μ

where vi, μ denotes electrical potential created by external electric field in the atomic orbital μ in the i -th unit cell. Moreover, Magnetic exchange field (MEF) in monolayer WSe2 has experimentally been observed with a ferromagnetic EuS substrate by Zhao et al. [16]. Having a great magnetic moment of Eu2 + (Sz ≈ 7μ B ) and also a great exchange coupling ( J≈ 10meV), EuS can create a great MEF (∝JSz) [17]. When we put a WSe2 monolayer on the EuS substrate, we will have an induced magnetic exchange field in the monolayer, caused by a magnetic proximity effect. The detailed explanations of proximity effect in ferromagnetic/semiconductor hybrids have been expressed by Korenev [18]. It should be mentioned that exchange interaction annihilates the time-reversal symmetry in that ribbon and can generate spin polarization. In the tight binding method, a magnetic exchange field can rise with adding a term to initial Hamiltonian. Therefore, the new Hamiltonian with an exchange field can be expressed as following:

where H01 is connection Hamiltonian between H0 and H1 unit cells, and Hm,m+1 is connection Hamiltonian between Hm and Hm+1 unit cells, (Fig. 2). Because the electrodes and device are the same material (WSe2 monolayer), there are no different dispersions at the boundaries. So H01 L and Hm·m+1 are equal. Notice that g00 and gmR+1·m+1 are left and right surface Green's function respectively. The details of calculating the surface Green's functions are provided in Appendix B. After calculating Hamiltonian of the system, the Green's function can be obtained as following:

G s = ((E+iη) − Hs)−1,

∑ [tij·μνc†i·μ cj·ν + H.C] + ∑ vi,μ c†i .μ ci .μ, ij·μν

(14)

denote the conductance for up and down spins, re-

161

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nanoribbon, Fig. 6(a). The occurrence of metallic phase in armchair WSe2 nanoribbon could be more because of varying the charge distribution in edges due to the transverse electric field. Variations of the charge distribution in lattice points can modify onsite energies which lead to change of the total Hamiltonian of system. Therefore, a metallic phase with the presence of external electric field is probable. Such a metal phase for armchair MoS2 nanoribbon in presence of an external transverse electric field has been reported by Dolui et al. [21]. Also, one can see that the spin degeneracy is broken after applying transverse electric field in armchair ribbon. Because applying the transverse electric field can change electric potential of atoms in the crystal and this modifies the onsite energies. Therefore, a small spin splitting is occurred. Using Green's function method, we also calculate total density V of states (TDOS) for armchair WSe2 nanoribbon in the presence of 2.1 w of transverse electric field, Fig. 6(b). In which the energy states close to Fermi energy is exist so that a metallic phase is obviously observed. The DOS plot is obviously corresponded to energy band structure in this situation, see Fig. 6. Notice that the picks indicated on TDOS plot represent Van Hove singularities [22]. Stark effect is described as a variations of energy gap as a function of external electric field, which could be either linear or nonlinear effect [23]. This effect has already been observed in Boron nitride nanoribbons and nanotubes [24]. The linear giant Stark effect is characterized with the following equation

Fig. 3. Energy band structure of WSe2 nanosheet. A direct band gap is occurred in K point.

diagonalizing Eq. (6) and results are shown in Fig. 4. As it is seen in Fig. 4(a), in armchair nanoribbon a semiconducting state with energy band gap of 0.71 eV is observed. Spin degeneracy is perceived because the spin up and spin down curves are overlapped, Fig. 4(a). But in zigzag WSe2 nanoribbon, spin up and down are separated so that red and blue bands indicate spin up and spin down states respectively, (see Fig. 4(b)). Moreover, the zigzag structure is in metallic phase for both spin components. An interesting result is emerging an energy gap of 0.23 eV with an applied transverse electric field of V 2.1 w upon zigzag nanoribbon, Fig. 5(b). While with applying transverse

Eg (E ) = Eg,0 + eSL E ,

(17)

where E is external electric field, e is electron charge and SL is the linear coefficient of the giant Stark effect. It should be mentioned that SL denote strength of external electric field to modify the energy band gap. As it is shown in Fig. 7, linear relations are occurred between variations of transverse electric field as a function of energy band gap for both armchair (a) and zigzag (b) nanoribbons. For armchair nanoribbon, we observe the linear giant stark effect V V with SL = −10.44 A0 in the range of 0.1 w to 2.1 w of applied transverse electric fields. While the calculated value of SL for zigzag ribbon in the V V range of − 1.8 w to − 1.4 w of transverse applied electric fields is 7.14 A0 . Furthermore, we peruse effect of a magnetic exchange field induced by a ferromagnetic EuS substrate upon both armchair and zigzag WSe2 nanoribbons. An interesting result with applied 0.2 eV of exchange field

V

electric field of − 2.1 w (only by reversing the direction of transverse electric field) upon the zigzag WSe2 nanoribbon a metallic phase is observed, see Fig. 5(a). This is because the zigzag WSe2 nanoribbon has two different edges named W and Se edges (see Fig. 2.). So that such geometry leads to difference between its band structures by reversing the direction of applied transverse electric field with equal values of it. Moreover, a semiconductor-metal phase transition is occurred with V 2.1 w of applied transverse electric field upon armchair WSe2

Fig. 4. Electronic structure of (a) armchair and (b) zigzag WSe2 nanoribbon. Red and blue bands refer to spin up and down, respectively. There exist n = 38 and n = 36 atoms in the unit cell for zigzag and armchair nanoribbons respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.) 162

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M. Shahbazi

V

V

Fig. 5. Energy dispersion curves of zigzagWSe2 nanoribbons with (a) − 2.1 w and (b) 2.1 w of transverse electric fields. Red and blue bands refer to spin up and down,

respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

V

Fig. 6. (a) Energy band structure and (b) Total density of states (TDOS) for armchair WSe2 nanoribbon in presence of transverse electric field of 2.1 w . Red and blue

bands refer to spin up and down, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Fig. 9(a). As it is anticipated, magnetic-induced exchange field cause spin polarization in armchair structure Fig. 9(b). As a result, by manipulation of chemical potential (which leads to change of Fermi energy) and by tuning the strength of external exchange field one can construct a desired spin filter device.

is a half-metallic state for armchair WSe2 nanoribbon in which spin up curve demonstrates metal phase while spin down curve exhibits semiconducting state with 0.73 eV of energy gap, see Fig. 8(a). In the case of zigzag WSe2 nanoribbon with an induced exchange field of 0.2 eV, splitting the bands with opposite spins is clearly observed so that Spin up and down bands are pushed to up and down, respectively, Fig. 8(b). We also calculate electronic conductance of armchair WSe2 nanoribbon in presence of 0.2 eV of magnetic exchange field induced by a ferromagnetic substrate using Green's function method,

6. Conclusion In this work, we have investigated the electronic structure of WSe2 163

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M. Shahbazi

Fig. 9. (a) Electronic conductance and (b) Spin polarization curves for armchair WSe2 nanoribbon in the presence of an indirect exchange field of 0.2 eV. Red and blue curves denote spin up and down, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.) Fig. 7. Variations of energy band gap as a function of transverse electric field for bothWSe2 (a) armchair and (b) zigzag nanoribbons.

We have also extracted the linear giant Stark effect for armchair and zigzag WSe2 nanoribbons by varying the external transverse electric field. Furthermore, we have studied an exchange interaction induced by a ferromagnetic EuS substrate with magnetic proximity effect. Spin polarization have also been generated in armchair WSe2 nanoribbon with an external exchange field.

monolayer. We consider six-band tight binding model which is invariant with respect to mirror symmetry of the system. A direct band gap of 1.74 eV have been observed for WSe2 nanosheet. Moreover, the semiconducting and metallic phases for the ground states of armchair and zigzag WSe2 nanoribbons have been apperceived, respectively. The electronic structure of WSe2 nanoribbons can be tuned by applying transverse electric field. Both semiconductor-metal phase transition and occurrence of an energy gap are the results of applying transverse electric field for armchair and zigzag WSe2 nanoribbons, respectively.

Acknowledgment I would like to thank Saeid Mollaei for his helps.

Fig. 8. Band structure of (a) armchair and (b) zigzag WSe2 nanoribbons in the presence of an exchange field of 0.2 eV. Red (blue) band refers to Spin up (down). Notice that Fermi energy is set to zero. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.) 164

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Appendix A In this appendix, both onsite and hopping energies for six-band tight binding model of WSe2 single-layer are provided. The onsite energies can be written as following:

ε 0⎤ ε=⎡M , ⎢ 0 εX ⎥ ⎣ ⎦

(A1)

0 0 ⎤ ⎡ f0 f2 − iλM SˆZ ⎥, ∈M = ⎢ 0 ⎥ ⎢ ⎢ 0 iλM SˆZ f2 ⎥ ⎦ ⎣

(A2)

λ

⊥ 0 ⎤ − i 2x SZ ⎡ fp + t xx ⎢ λ ⎥ X ⊥ ∈X = ⎢ i SZ fp + t yy 0 ⎥, 2 ⎢ ⎥ ⊥ 0 0 fz − tzz ⎢ ⎥ ⎣ ⎦

(A3) ⊥ that t xx

where λM = 0.241 eV and λX = 0.439 eV denote the spin orbit coupling for Tungsten and Selenium atoms, respectively. Notice ⊥ tzz = Vppσ. Furthermore, the matrices of hopping energies for both nearest and next nearest neighbors are indicated as following

t1MX =

t2MX =

t3MX =

t1MM =

(A4)

− 6 3 Vpdπ + 2Vpdσ 12Vpdπ + 3 Vpdσ ⎤ ⎡ 0 2 ⎢ , 0 − 6Vpdπ − 4 3 Vpdσ 4 3 Vpdπ − 6Vpdσ ⎥ ⎥ 7 7⎢ ⎢14Vpdπ ⎥ 0 0 ⎣ ⎦

(A5)

3 3 Vpdπ − Vpdσ 12Vpdπ + 3 Vpdσ ⎤ ⎡ 9Vpdπ − 3 Vpdσ 2 ⎢ − − − − 5 3 V 3 V 9 V 3 V 2 3 Vpdπ + 3Vpdσ ⎥ pdπ pdσ pdπ pdσ ⎥, 7 7⎢ ⎢ − Vpdπ − 3 3 Vpdσ − 5 3 Vpdπ − 3Vpdσ − 6Vpdπ + 3 3 Vpdσ ⎥ ⎣ ⎦

(A6)

3Vddδ + Vddσ 1⎡ ⎢ 3 (V − V ) ddδ ddσ 4⎢ 0 ⎣

(A7)

⎡ 1⎢ MM t3 = ⎢ 4⎢ ⎢ ⎣

3 (Vddδ − Vddσ ) Vddδ + 3Vddσ 0 3 2

3 4

(−Vddδ + Vddσ )

3 (Vddδ 2

− Vddσ )

⎤ ⎥ (Vddδ − 4Vddπ + 3Vddσ ) ⎥, + 12Vddπ + 3Vddσ ) ⎥ 1 (Vddδ − 4Vddπ + 3Vddσ ) 4 (3Vddδ + 4Vddπ + 9Vddσ ) ⎥ ⎦ 3 2

3 4

− 2 (Vddδ − Vddσ )

3 4

3 (Vddδ 2

⎤ ⎥ (Vddδ − 4Vddπ + 3Vddσ ) ⎥, + 12Vddπ + 3Vddσ ) − ⎥ 1 (Vddδ − 4Vddπ + 3Vddσ ) 4 (3Vddδ + 4Vddπ + 9Vddσ ) ⎥ ⎦ (−Vddδ + Vddσ )

⎡ 3Vppπ + Vppσ 1⎢ 3 (Vppπ − Vppσ ) 4⎢ ⎢ 0 ⎣

(A8)

− Vddσ )

3 4

1 (Vddδ 4

−

3

(−Vddδ + Vddσ )

1 (Vddδ 4

3Vddδ + Vddσ 3 2

0 ⎤ 0 ⎥, ⎥ 4Vddπ ⎦

0 ⎤ ⎡Vppσ 0 t1XX = ⎢ 0 Vppπ 0 ⎥, ⎢ ⎥ 0 0 Vppπ ⎣ ⎦

t3XX =

= Vppπ and

⎡− 9Vpdπ + 3 Vpdσ 3 3 Vpdπ − Vpdσ 12Vpdπ + 3 Vpdσ ⎤ 2 ⎢ 5 3 Vpdπ + 3Vpdσ 9Vpdπ − 3 Vpdσ − 2 3 Vpdπ + 3Vpdσ ⎥ ⎥, 7 7⎢ ⎢− Vpdπ − 3 3 Vpdσ 5 3 Vpdπ + 3Vpdσ 6Vpdπ − 3 3 Vpdσ ⎥ ⎣ ⎦

⎡ 3Vddδ + Vddσ 1⎢ 3 MM t2 = ⎢ (−Vddδ + Vddσ ) 4⎢ 2 ⎢ − 3 (V − V ) ddσ ⎣ 2 ddδ

t2XX =

=

⊥ t yy

(A9)

(A10)

3 (Vppπ − Vppσ ) Vppπ + 3Vppσ 0

0 ⎤ , 0 ⎥ ⎥ 4Vppπ ⎥ ⎦

(A11)

− 3 (Vppπ − Vppσ ) 3Vppπ + Vppσ 0 ⎤ ⎡ 1⎢ − 3 (Vppπ − Vppσ ) , + V 3 V 0 ⎥ ppπ ppσ ⎥ 4⎢ ⎢ 0 0 4Vppπ ⎥ ⎣ ⎦

(A12)

This equations with more details can be found in Ref. [25]. Appendix B In this appendix, the details of calculation of surface Green's functions are described. This surface Green's functions are defined as following 165

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∼ L g00 = [(E+ iη) I − H00 − H−† 10 T],

(B1)

gmR+1.m+ 1 = [(E+ iη) I − Hm+1.m+ 1 − Hm.m+ 1T]−1,

(B2) ∼ where, η is very small number close to zero, which is utilized for eliminate singularity. Notice that T and T are transfer matrices and are calculated as:

T= t 0 + ˜t 0t1 + ˜t 0˜t1t2 + …,

(B3)

∼ ˜ T = t 0 + t 0˜t1 + t 0t1˜t2 + …,

(B4)

where t and ˜t are Green's function matrices, calculated from following recurrence equations

t 0 = [(E+ iη) I − H00]−1H−† 10,

(B5)

˜t 0 = [(E+ iη) I − H00

(B6)

]−1H

−10,

t i = (I − t i − 1˜t i − 1 − ˜t i − 1t i − 1)−1t i2− 1,

(B7)

˜t i − 1ti − 1)−1˜t i2− 1,

(B8)

˜t i = (I − t i − 1˜t i − 1 −

This calculations with more details can be found in Ref. [19].

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