quantum transport model for zigzag molybdenum disulfide nanoribbon structures a full quantum framework

Quantum transport model for zigzag molybdenum disulfide nanoribbon structures : A full quantum framework , Chun-Nan Chen , Feng-Lin Shyu, Hsien-Ching Chung, Chiun-Yan Lin, and Jhao-Ying Wu Citation: AIP Advances 6, 085123 (2016); doi: 10.1063/1.4962346 View online: http://dx.doi.org/10.1063/1.4962346 View Table of Contents: http://aip.scitation.org/toc/adv/6/8 Published by the American Institute of Physics AIP ADVANCES 6, 085123 (2016) Quantum transport model for zigzag molybdenum disulfide nanoribbon structures : A full quantum framework Chun-Nan Chen,1,a Feng-Lin Shyu,2 Hsien-Ching Chung,3 Chiun-Yan Lin,3 and Jhao-Ying Wu4 Quantum Engineering Laboratory, Department of Physics, Tamkang University, Tamsui, New Taipei 25137, Taiwan Department of Physics, R.O.C Military Academy, Kaohsiung 830, Taiwan Department of Physics, National Cheng Kung University, Tainan 70101, Taiwan Center of General Studies, National Kaohsiung Marine University, Kaohsiung 811, Taiwan (Received 19 May 2016; accepted 25 August 2016; published online 31 August 2016) Mainly based on non-equilibrium Green’s function technique in combination with the three-band model, a full atomistic-scale and full quantum method for solving quantum transport problems of a zigzag-edge molybdenum disulfide nanoribbon (zMoSNR) structure is proposed here For transport calculations, the relational expressions of a zMoSNR crystalline solid and its whole device structure are derived in detail and in its integrity By adopting the complex-band structure method, the boundary treatment of this open boundary system within the non-equilibrium Green’s function framework is so straightforward and quite sophisticated The transmission function, conductance, and density of states of zMoSNR devices are calculated using the proposed method The important findings in zMoSNR devices such as conductance quantization, van Hove singularities in the density of states, and contact interaction on channel are presented and explored in detail C 2016 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/) [http://dx.doi.org/10.1063/1.4962346] I INTRODUCTION The group-VIB transition-metal dichalcogenides with a chemical formula MX (M = Mo,W and X = S, Se, Te), a recently emerged member of the two-dimensional (2D) layered crystal family, are constructed by stacking multiple X - M - X layers.1–3 The MX family has recently attracted tremendous research because of the existence of a suitable band gap and high electron mobility.1,3–5 Therefore, the natures of the MX family produce their potential application as the next-generation channel material for field-effect transistors.6–8 A zigzag-edge molybdenum disulfide (MoS2) nanoribbon, which is abbreviated to zMoSNR, has been used as the channel material of field-effect transistors.9,10 The quasi-1D semiconductor natures of zMoSNR transistors create an outstanding advantage of constructing transistors such as excellent mobility, low power dissipation, high on/off ratio, and high density package, etc.8,10,11 In monolayer MoS2, conduction and valence bandedges are mainly dominated by d orbitals (d z 2, d x y , and d x 2−y 2) of Mo atom, which have band extreme located at the Brillouin-zone corners K+ and K−.1,12–14 In this study, a compact but practical model, three-band tight-binding model in the Mo - d z 2, d x y , d x 2−y basis given recently by Liu et al.,1 is adopted as one of our study foundations By adopting only the nearest-neighbor Mo-Mo interactions, this three-band model can exactly describes the physical natures of MoS2 monolayer and nanoribbon near the bandedge region.1,14 Therefore, the three-band model for MoS2 monolayer and nanoribbon studies can yields satisfactory results, because the physic natures near bandedge are the focus of our research a Electronic mail: quantum@mail.tku.edu.tw & ccn1114@kimo.com 2158-3226/2016/6(8)/085123/15 6, 085123-1 © Author(s) 2016 085123-2 Chen et al AIP Advances 6, 085123 (2016) To describe electronic transport through a zMoSNR device, we perform ballistic quantum transport simulations within the non-equilibrium Green’s function (NEGF) formalism15–19 using a nearest-neighbor three-band model.1 The basic idea of this proposed method is to use a minimum number of orbitals and parameters to describe, as accurately as possible, the most relevant portion of the energy-band diagram of a zMoSNR Previously, many investigations had displayed that Green’s functions could be obtained by the iterative method.20–22 Iterative method may be suitable to finish this job or not (divergent), but the major drawback of this method is that a relatively slow convergence is emerged in the self-consistent procedure Moreover, the other calculating method for fulfilling the Green’s functions is based on the Dyson equation treatment developed by Caroli et al and subsequently used by a number of theoretical researchers.23–25 The major drawback of the Dyson approach is that the boundary conditions become extremely complicated to solve when many bands are involved However, the drawbacks of the iterative method and Dyson approach could be shunt in this study by the proposed non-iterative method, a NEGF framework in combination with the complex energy-band method.26–28 The aim of this proposed method is to develop a straightforward but sophisticated NEGF technique for solving those quantum transport problems of a zMoSNR Consequently, the method developed in this paper is expected to be widely adopted due to its conceptual simplicity, computational efficiency, and applied versatility In this paper, by using the proposed method we present a comprehensive investigation on the physical properties of zMosNR devices, such as transmission function, conductance, and density of states (DOS) One of our important findings in zMoSNR devices is the existence of conductance quantization29–33 in a short and narrow flat-band zMoSNR structure connecting two ballistic 2D contacts Other important finding of zMoSNR devices is the van Hove singularities (vHSs)30,34–37 in the DOS because of the discontinuous DOS phenomena In this paper, we impose the potential profile of double-barrier structures (DBSs) on zMoSNR devices, and we explore the isolation effect of the DBS barrier potential on the conductance quantization and DOS spectrum of zMoSNR devices The vHSs in the DOS produce the sharp peak at vHS points for some 1D or 2D crystalline solids, which result in strong influences on their physical behaviors.30,34–37 The vHSs in the 1D or 2D crystalline solids have been found in carbon nanotube, twisted bilayer graphene, and monolayer silicene under uniaxial strain.30,34–37 II THEORETICAL METHOD Here we briefly outline the basic structure of our sample, as shown in Fig We wish to compute the conductance, transmission function, and DOS for a zMoSNR with a central channel region of interest consisting of l atomic layers labeled σ = 1, 2, · · ·, l We assume that flat-band conditions exist in the left incoming (σ ≤ 0) and right outgoing (σ ≥ l + 1) contacts outside the central channel region of l atomic layers Basically, the used three-band model1 of monolayer MoS2 is a tight-binding scheme with d -like (d = d z 2, d x y , d x 2−y 2) unit-cell-scale basis orbital The three-band matrix elements of monolayer MoS2 Hamiltonian are given as shown in Appendix A.1 FIG Geometric structure of a zMoSNR-based transport device (width : N zigzag lines; channel length: l atomic layers) 085123-3 Chen et al AIP Advances 6, 085123 (2016) The state function |k⊥⟩ of a zMoSNR with N zigzag lines (width) in the flat-band region is a linear superposition of the × N terms of tight-binding Bloch basis functions |k⊥, j, d >, which can be expressed as |k⊥ > = N   b j, d (k⊥)|k ⊥, j, d >, (1) j=1 d=1 where b j, d is the linear expansion coefficient, d denotes the symmetry-type d -like orbital, j specifies the in-layer (∥) lattice site of N atoms of Mo within the zMoSNR unit cell, and k⊥ is the wave vector along the zMoSNR channel direction (⊥) Note that the electron energy E of |k⊥, E > and b j, d (k⊥, E) is abbreviated for conciseness here Moreover, the tight-binding Bloch basis functions can be written as  |k⊥, j, d > = √ exp(ik ⊥σa ′)|σ, j, d >, (2) Ω σ where Ω is the normalization factor, σ is an integer layer label (see Fig 1), a ′ is the spacing between two adjacent layers, and |σ, j, d > is the d -like orbital of Mo atom at site (σ, j) Based on the |k⊥, j, d > basis, the zMoSNR Hamiltonian HzMoSNR(k⊥), which possess 3N × 3N matrix form, is shown in Appendix B.1 Meanwhile, the state function of a zMoSNR can also be written in the form as  |k ⊥ > = cσ, j, d (k ⊥)|σ, j, d >, (3) σ j, d where cσ, j, d (k⊥) = √1Ω b j, d (k ⊥) exp(ik ⊥σa ′) Therefore, the Hamiltonian of a zMoSNR can also be expressed in terms as HzMoSNR(k⊥) = Hσ,σ−2e−ik⊥2a + Hσ,σ−1e−ik⊥a + Hσ,σ + Hσ,σ+1e+ik⊥a + Hσ,σ+2e+ik⊥2a , (4) ′ ′ ′ ′ where Hσ,σ and Hσ,σ ′ are 3N × 3N matrices (see Appendix C) whose elements are given by (Hσ,σ ) j, d, j ′, d′ = < σ, j, d|(H − E)|σ, j ′, d ′ > (5a) (Hσ,σ ′) j, d, j ′, d′ = < σ, j, d|H |σ ′, j ′, d ′ >, (5b) and respectively, and σ ′ denote σ ± or σ ± The Schrödinger equation (H − E)|k⊥ > = in a flat-band zMoSNR can be written in the transfer-matrix form as26–28,38–41 −H −1 Hσ,σ−1 −H −1 Hσ,σ −H −1 Hσ,σ+1 −H −1 Hσ,σ+2 cσ−1 cσ−1 σ,σ−2 σ,σ−2 σ,σ−2  σ,σ−2      I 0    cσ  = e−ik⊥a′  cσ  ,   cσ+1 cσ+1 I 0       0 I   cσ+2 cσ+2 (6) where I is a 3N × 3N identity matrix, a state function |k ⊥⟩ is the available plane-wave states at the flat-band region on the left and right contacts, and cσ denotes a 3N-length column vector of coefficients whose components are cσ, j, d , i.e.,  cσ,1     cσ,2  cσ =       cσ,3N  (7) ′ By Bloch’s theorem, the tight-binding coefficients must obey the relation cσ = e−ik⊥a cσ+1 085123-4 Chen et al AIP Advances 6, 085123 (2016) In the whole structure with perturbation, the wave function of the Schrödinger equation (H − E)|ψ > = can be expressed as   |ψ > = a(k⊥)|k⊥ > = a(k ⊥)cσ, j, d (k ⊥)|σ, j, d >, (8) σ k⊥ j, d k ⊥ where a(k⊥) is the corresponding amplitude coefficient of a zMoSNR state function |k ⊥⟩ Therefore, the wave function of the whole structure may also be written as  |ψ > = f σ, j, d |σ, j, d >, (9) σ where f σ, j, d denotes  k⊥ j, d a(k⊥)cσ, j, d (k⊥), which is electron-energy dependent, i.e., f σ, j, d (E) Writing the Schrödinger equation (H − E)|ψ > = in the |σ, j, d > basis form, we obtain a linear equation situated at the σth layer as27,39–41 Hσ,σ−2 f σ−2 + Hσ,σ−1 f σ−1 + Hσ,σ f σ + Hσ,σ+1 f σ+1 + Hσ,σ+2 f σ+2 = 0, (10) where Hσ,σ and Hσ,σ ′ are 3N × 3N matrices (see Appendix C) and f σ denotes a 3N-length column vector whose components are f σ, j, d Therefore, the Hamiltonian matrix of the whole structure can be written as     ˆ H − E I =  · · ·   · · ·   Hσ−1,σ−3 Hσ−1,σ−2 Hσ−1,σ−1 Hσ−1,σ ··· ··· Hσ−1,σ+1 ··· Hσ,σ−2 Hσ,σ−1 Hσ,σ Hσ,σ+1 Hσ,σ+2 ··· Hσ+1,σ−1 Hσ+1,σ Hσ+1,σ+1 Hσ+1,σ+2 Hσ+1,σ+3 ··· ···    · · ·   · · ·       (11) For a given electron energy E, solving Eq (6) yields a set of × (3N) real or complex wave vector k⊥, λ ; λ = 1, 2, , 12N and their associated state functions k⊥, λ which can results in a E − k⊥, λ=1∼12N complex-band structure.26–28 We reorder the state functions such that λ = 1, 2, , 6N corresponds to states which either propagate (k⊥ real) or decay (k⊥ complex) to the right,26,27,40,41 while λ = 6N + 1, 6N + 2, , 12N corresponds to those which either propagate or decay to the left The boundary conditions are such that we have a known incoming plane-wave state from the left contact, no incoming from the right contact, and unknown outgoing transmitted and reflected plane-wave states in the right and left contacts, respectively Proper open boundary conditions at the two contacts can be obtained as a linear combination of complex-band structure solutions For a given energy E and for a given amplitude (here unitary) of an incoming plane-wave-like state (denoted by i) from the left, the wave functions in the left (L) and right (R) contacts must fulfill the boundary conditions of this problem, which can be described in the state functions of complex-band structure as follows:26,27,40,41 |ψ; L > = |ψ i > +|ψ ℜ > = Ii |k⊥,i ; L > + 6N  a(k⊥, λ+6N ; L)|k ⊥, λ+6N ; L > (12a) λ=1 and |ψ; R > = |ψ ℑ > = 6N  a(k⊥, λ ; R)|k ⊥, λ ; R >, (12b) λ=1 where Ii represents the known amplitude (here unitary) coefficient of the incoming plane-wave-like state function from the left contact, and a(k ⊥, λ ; R) and a(k⊥, λ+6N ; L) are the unknown amplitude coefficients for the transmitted and reflected state functions, respectively For convenience, we 085123-5 Chen et al AIP Advances 6, 085123 (2016) denote ℜ and ℑ to represent the outgoing waves which propagate (or decay) to the left in the left contact and right in the right contact, respectively  ζ Based on Eq (9), we rewrite it in the form as |ψ ζ > = f σ, j, d |σ, j, d >, then we obtain the σ j, d relation as follows:41 ζ  f ζ   B ζ  a ζ  Bσ,2  ζσ  =  ζσ,1   1ζ  , ζ  f σ+1  Bσ+1,1 Bσ+1,2  a2  where ζ denote ℑ and ℜ for transmitted and reflected waves, respectively, (13) ℑ ℑ [Bσ,1 Bσ,2 ] = [cσ,1cσ,2 · · · cσ,6N ] R , (14a) ℜ ℜ [Bσ,1 Bσ,2 ] = [cσ,6N +1cσ,6N +2 · · · cσ,12N ] L , (14b)  a(k ⊥,1; R)     a(k ⊥,2; R)  ℑ ℑ a1 =   and a2 =    a(k⊥,3N ; R) a(k⊥,3N +1; R)   a(k⊥,3N +2; R)   ,    a(k⊥,6N ; R)  (15a) and a(k ⊥,6N +1; L)   a(k ⊥,6N +2; L) ℜ ℜ a1 =   and a2 =     a(k ; L) ⊥,9N  a(k⊥,9N +1; L)   a(k⊥,9N +2; L)      a(k⊥,12N ; L)  (15b) The coefficients f σ, j, d of d -like orbitals in the right and left contacts can be obtained from the Eqs (13)-(15), which result in two boundary conditions in matrix form as15–18,41  f ℑ   B ℑ  l+3 =  l+3,1 ℑ ℑ  f l+4   Bl+4,1  ℑ Bl+3,2  ℑ Bl+4,2  Bℑ  l+1,1 ℑ  Bl+2,1  −1  f ℑ  ℑ Bl+1,2   l+1 ℑ ℑ   f l+2  Bl+2,2 (16a) and  f ℜ   B ℜ ℜ   ℜ ℜ  −1  ℜ   −3 =  −3,1 B−3,2  B−1,1 B−1,2  f −1 (16b) ℜ ℜ ℜ ℜ ℜ  f −2   B−2,1   B0,1   f 0ℜ B−2,2 B0,2 Combining the Eqs (11), (12), and (16), we can obtain the Schrödinger-like equation in the NEGF form within the active region (−1 ≤ σ ≤ l + 2) as15–19,41 Hact − E Iˆ + Σ L + Σ R {ϕ} = {S} , (17) where the Hamiltonian of active region (Hact), the boundary self-energies for the left (L) and right (R) regions (Σ L, R ), the wave function {ϕ}, and the source term {S} are  H−1,−1   H0,−1   H1,−1  Hact − E Iˆ =        H−1,0 H−1,1 0 H0,0 H0,1 H0,2 ··· H1,0 H1,1 H1,2 H1,3 0 Hl,l−2 Hl,l−1 Hl,l Hl,l+1 0 Hl+1,l−1 Hl+1,l Hl+2,l Hl+1,l+1 Hl+2,l+1 ···        ,  Hl,l+2   Hl+1,l+2  Hl+2,l+2 (18) 085123-6 Chen et al AIP Advances 6, 085123 (2016)  H  −1,−3 H−1,−2   ΣL =   H0,−2   H  l+1,l+3 Σ R =   Hl+2,l+3 Hl+2,l+4  Bℜ ℜ   ℜ ℜ  −1  −3,1 B−3,2  B−1,1 B−1,2 , ℜ ℜ ℜ ℜ  B−2,1   B0,1  B−2,2 B0,2  −1  ℑ  Bℑ ℑ ℑ  l+3,1 Bl+3,2  Bl+1,1 Bl+1,2 , ℑ ℑ ℑ ℑ    Bl+2,1  Bl+4,1 Bl+2,2 Bl+4,2 (19a) (19b) ℜ  f −1  ℜ   f   f    {ϕ} =   ,    f l   f ℑ    l+1 ℑ  f l+2 (20)    − H−1,σ cσ, i   σ=−3      − H0,σ cσ,i   σ=−2      H1,σ cσ,i  {S} =  −   σ=−1   −H2,0c0,i            (21) and The Green’s function of the device is simply defined as G d = E Iˆ − Hact − Σ L − Σ R −1 (22) Within the NEGF formalism: once the Green’s function G d is found, the transmission function T(E) is followed by the trace of18,42 T(E) = Tr[ΓL G d ΓRG+d ], (23) where T(E) denotes the product of the number of forward propagating eigenstates M(E), which is sometimes called as propagating channels or propagating modes, and the transmission probability T(E), Tr is the trace operator, ΓL, R = i(Σ L, R − Σ+L, R ) denote the broadening factors, and superscript ‘+’ denotes conjugate transpose According to Landauer formula, at zero temperature the linear response conductance G(E) of the system is evaluated at the Fermi energy EF , which can be related to the transmission function T(E) as18,31,42–44 G(EF ) = 2e2 T(EF ), h (24) where e is the electron charge, h is the Planck constant, 2e2/h is the conductance quantum, and the transmission function T(E) should be calculated at the Fermi energy EF , i.e., T(EF ) = Tr[ΓL G d ΓRG+d ]| E=E F 085123-7 Chen et al AIP Advances 6, 085123 (2016) With the Green’s function G d specified, the DOS can be obtained via18,30,42 DOS(E) = Tr[G d (ΓL + ΓR )G+d ] 2π (25) For concise expression and highlighting, the proposed method ignores the spin-orbit coupling effect in the present study However, after finishing the proposed method the spin-orbit coupling effect could be easily included by means of the Eq (27) of Ref III RESULTS AND DISCUSSION The energy-band diagrams of 2D transition-metal dichalcogenides are primarily determined by their crystal structure Structurally, monolayer MoS2 can be regarded as a tri-layer S - Mo - S sandwich, where one Mo layer alone produces a trigonal prismatic structure with two S layers Within each atomic layer, Mo and S atoms form 2D hexagonal lattices (six neighbors) When viewed from the top, it displays a honeycomb structure like graphene, with the A-sublattice being the Mo atom per site and B-sublattice being the two S atoms per site Therefore, monolayer MoS2 possess the D3h (C3, σh , and σ v ) point-group symmetry The electron configurations of the Mo and S atoms are Mo : [Kr] 5s14d and S : [Ne] 3s23p4 The Mo 4d- and S 3p-orbitals possess the outer shell of the electron configuration with the higher energy The Mo 4d- and S 3p- valence electrons mainly constitute the ion bond inside the trigonal prismatic structure, which result in the dominant effects on the physical properties of 2D layered MoS2 In addition to the symmetry of monolayer MoS2 2D hexagonal lattice (D6h ), it is important to include the intracell symmetry, especially in the local symmetry inside the trigonal prismatic structure (C3h ), which have important effects on the physic properties of monolayer MoS2 Monolayer MoS2 structure symmetry indicates that the Mo - 4d and S - 3p orbitals can be classified as three and two groups, respectively, as follows: {d z 2}, {d x 2−y 2, d x y }, {d x z , d y z } and {px , p y }, {pz }.1,13,14 Moreover, monolayer MoS2 D3h symmetry yields the hybridization of {d z 2} and {d x 2−y 2, d x y } near the bandedge regions which splits a band gap at this K± points First-principle calculations also proved that the bandedge states of monolayer MoS2, which are located at the K± points, are constituted by hybridization of the Mo - {d z 2}, {d x 2−y 2, d x y } orbitals with a little contribution of the S {px , p y } orbitals.1,13,14 Therefore, it is reasonable to only adopt three orbitals (Mo - d z 2, d x y , d x 2−y 2) for a compact tight-binding framework as basis Although the three-band model, which was given recently by Liu et al, only adopts three Mo - d orbitals as basis, yet these D3h symmetry-based d orbitals are actually the unit-cell-scale d -like basis of MoS2 which are constituted of not the pure D6h Mo - d orbitals but the hybridization with some S - p orbitals In a different point of view, the d -like orbitals of the three-band model are located at the center of the trigonal prismatic structure, and these D3h symmetry-based d - d hoppings account for not only the 2D hexagonal intercell d - d interaction of the Mo atoms but also the intracell interaction of Mo - d and S - p orbitals Namely, the three-band model takes the local symmetry C3h inside the trigonal prismatic structure into account, and hence the intracell symmetry and composition of the Mo - S ion bonds are implied into this model Although the three-band model is a three Mo - d orbital model, yet beyond the traditional Slater-Koster framework it implies the S - p orbitals inside Therefore, the three-band model can exactly describe near the bandedge properties of MoS2 monolayer and nanoribbon The three-band model reproduces a D3h point group symmetry rather than the oversymmetric D6h of the 2D hexagonal lattices, which includes the effects of microscopic symmetry within the unit cell The correct symmetry (D3h ) in monolayer MoS2 can be restored in the three d-orbital model (D6h ) by 12 22 12 12 adding E11 - and E12 -included terms [i.e., the imaginary terms: i2E11 sin α cos β, i2E11 sin(2α), √ 12 22 i2 3E11 cos α sin β, and i4E12 sin α(cos α − cos β)] to the × 2D hexagonal Hamiltonian matrix 12 22 of the three-band model, as shown in Appendix A In Eq (A1), the E11 - and E12 -included terms are the only matrix elements which are incompatible with the oversymmetric D6h , and those terms allows the inherent reduced symmetry inside the intracell to be evaluated Namely, the three-band 085123-8 Chen et al AIP Advances 6, 085123 (2016) FIG Energy-band diagram for monolayer MoS2 using the tight-binding three-band model in the nearest-neighbor approach model enables the effects of local perturbations to be described both qualitatively and quantitatively on a scale smaller than the unit cell As shown in Fig 2, the energy-band diagrams of monolayer MoS2, which is calculated by the tight-binding three-band model in the nearest-neighbor approach, agree well with the first-principle ones only for the valence- and conduction-bands in the K± valleys (see Fig 3(a) of Ref 1).1 Therefore, the used three-band model is able to capture the essential physics of monolayer MoS2 in the K± valleys reasonably well A zMoSNR with N zigzag lines owns N molybdenum atoms and 2N sulfide atoms inside its unit cell According to the three-band model in the Mo - d z 2, d x y , d x 2−y basis, there are N-center tight-binding framework, which results in a 3N × 3N dimensional Hamiltonian matrix, as shown in Appendix B The energy-band diagram of a zMoSNR with zigzag lines is calculated and shown in the left panel of Fig 3, and its corresponding number of forward propagating channels M(E) at a given energy E is shown in the right panel of Fig Moreover, the 3N × 3N three-band model in Appendix B can give reasonable results in the conduction bandedge portion, as shown in Fig 3, which are verified with those of the first-principle calculation (see Fig of Ref 1).1 Note that the conduction bandedge portion of a zMoSNR is the focus of our study ′ ′ By solving the eigenvalue (e−ik⊥a ) of Eq (6), we can obtain a set of 3N × real (if |e−ik⊥a | −ik ⊥a ′ = 1) or complex (if |e | 1) wave vectors k⊥ for a given E The given E and its associated real k⊥ produce the conventional E-k ⊥ energy-band diagram of a zMoSNR with zigzag lines, as shown in the left panel of Fig The real k⊥ yields the propagating waves which propagate to the right (υg (k⊥) > 0) or left (υg (k⊥) < 0) direction (see Fig 3), where υg denotes the group velocity ( −1∂H N R B(k ⊥)/∂k⊥), while the complex k ⊥ yields the evanescent waves which decay exponentially to the right (Im(k⊥) > 0) or left (Im(k⊥) < 0) direction Each of the eigenvectors of Eq (6) corresponds to a pair of k⊥ and −k ⊥, and hence we have one-half (6N) rightward and the other-half (6N) leftward propagating or evanescent waves for a given energy E, as shown in Fig To match the boundary conditions, all state functions (12N) including the evanescent states must be taken into account as shown in Eqs (12a) and (12b) For this purpose, we reorder the state functions such that λ = 1, · · ·, 6N correspond to states which either propagate or decay to the right, while λ = 6N + 1, · · ·, 12N correspond to those which either propagate or decay to the left Using the open boundary conditions (Σ) to take the contact interaction into account, we can convert the infinitely-dimensional Hamiltonian matrix (see Eq (11)) of the entire device into the finitely-dimensional Hamiltonian matrix (see Eqs (18) and (19)) only computing the active region of device with fully reflecting boundary conditions.18,42 This term [Hact − E + Σ L + Σ R ] in Eq (17) 085123-9 Chen et al AIP Advances 6, 085123 (2016) is not Hermitian, because the boundary self-energies Σ L, R are not Hermitian matrices Consequently, the non-Hermitian self-energies produce that the number of electrons in the channel region of device is not conserved Therefore, the Schrödinger-like equation in the NEGF form, as shown in Eq (17), is constituted of the usual Schrödinger equation and two incremental terms ([Σ L + Σ R ] {ϕ} and {S}) Namely, E {ϕ} = Hact {ϕ} + [Σ L + Σ R ] {ϕ} − {S},18,42 where [Σ L + Σ R ] {ϕ} denotes an electron outflow from the channel region to the left and right contacts, {S} denotes an electron inflow from the left (i.e., external-source) contact to the channel region, Hact describes the physic characteristics of the active region of device, and E is the injecting electron energy from an external source According to the Schrödinger-like equation in the NEGF form, the wave function can be written as {ϕ} = (E − Hact − Σ L − Σ R )−1 {S} Furthermore, by definition of the NEGF framework, a wave function {ϕ} is the Green’s-function response of a source term {S}.18,42 Namely, {ϕ} = G d {S} Consequently, the Green’s function in the active region can be written as G d = (E − Hact − Σ L − Σ R )−1, as shown in Eq (22), where the boundary self-energies Σ L(R) describe the interaction of the left (right) contacts to the central channel region of device Therefore, the Green’s function G d describes the dynamic behavior of the electrons inside the active region of device and the interaction of the left and right contacts to the channel region The imaginary part of self-energies Im(Σ) produces the result of offering the finite lifetime (τ) of electrons at the device eigenstates, which gives rise to the escape rate (1/τ) of electrons through the contacts.18,42 Owing to Γ = −2Im(Σ), the broadening factor Γ dominates the exchange rate at which electrons can escape through the contacts With the Green’s function G d specified, it is easily understood that the transmission function T(E) of the entire device can be expressed in the framework as T(E) = Tr ΓL G d ΓRG+d , as shown in Eq (23) Moreover, the broadening factor Γ determines the coupling strength of the contacts to the channel region and the spreading of the contact states into the channel region, which produces the broadening of the channel states from its initial discrete levels to a continuous DOS spectrum.18,42 With the Green’s function G d specified, it is easily understood that the DOS inside the active region of device can be expressed in the framework as DOS(E) = 2π Tr[G d (ΓL + ΓR )G+d ], as shown in Eq (25) Figures 4(a)-4(d) display the diagrams of conductance (G), transmission function (T), and density of states (DOS) spectrum as a function of electron energy E for a zMoSNR device (width N = zigzag lines; channel length l = 28 atomic layers) A 2-24-2 atomic-layer DBS potential profile with barrier height 0.0 (i.e., flat-band), 0.1, 0.3, and 0.7 eV are imposed on this zMoSNR device, as shown in Figs 4(a)-4(d), respectively To demonstrate the validity of a proposed quantum transport model, a good first step is the calculation of conductance, transmission function, and DOS, which results in a reasonable result Figures 4(b)-4(d) compare DOS(E) with T(E) using the FIG Energy-band diagram for a zigzag-edge MoS2 nanoribbon with zigzag lines and its corresponding number of propagating channels at a given energy E 085123-10 Chen et al AIP Advances 6, 085123 (2016) FIG Transmission function (T ), conductance (G), and density of states (DOS) as a function of electron energy E for a zMoSNR device (width N = zigzag lines; channel length l = 28 atomic layers) A 2-24-2 atomic-layer DBS potential profile with barrier height (a) 0.0 (i.e., flat-band), (b) 0.1, (c) 0.3, and (d) 0.7 eV are imposed on this zMoSNR device, where M(E) denotes the number of propagating channels at a given energy E proposed method, which yield the very good agreements of results between the two data of DOS(E) and T(E) Moreover, the DOS spectrum also shows a similar behavior as the conductance spectrum, as shown in Figs 4(b)-4(d) Note that the anti-crossing behavior of subbands at E = 0.72 eV, as shown in Fig 3, leads to the dramatic DOS and conductance dips at this special energy position 085123-11 Chen et al AIP Advances 6, 085123 (2016) E = 0.72 eV, as shown in Figs 4(a)-4(d), which originates from the low symmetry of the material structure As shown in Fig 4(a), discrete conductance steps are obviously visible in a zMoSNR conductor with perfect flat-band potential (where T(E) = 1), which is known as conductance quantization in units of 2e2/h.29–33 According to Eq (24), the conductance of a zMoSNR with perfect flat-band potential (where T(E) = M(E)) provides a demonstration of the staircase conductance, which is the number of propagating channels M(E) multiplied by the conductance quantum 2e2/h, as shown in the right panel of Fig and Fig 4(a) In addition to the conductance step 4e2/h, some high conductance steps with values 4ne2/h (where n = 2, 3, 4, ) also emerge These high conductance steps originate from the increasing channels of zMoSNR conduction bands As shown in Figs 4(a)-4(d), we find the evolution of quantized conductance into resonant-tunneling conductance in DBSs When the barrier height of DBSs is raised enough, all quantum conductance steps are destroyed and the conductance is reduced, as shown in Fig 4(d) One of the physical properties for 1D crystalline solid is that their DOS produces a discontinuous function of electron energy, and it decreases exponentially and then increases abruptly in a discontinuous peak These sharp and asymmetric peaks shown in 1D crystalline solid are so-called vHSs30,34–37 A vHS is a non-smooth phenomenon in the DOS spectrum For a diagram of the DOS versus electron energy E, the vHSs happen where the derivative of DOS with respect to energy E is divergent, which will yield the strikingly sharp peak in the DOS spectrum When there are some critical points in the electron energy-momentum diagram given |∇k Ek | = 0, they can be called as vHS points Namely, |∇k Ek | = is a necessary condition for a vHS, which results in the singularity phenomenon in the DOS Similarly, as shown in Fig 4(a), in a perfect flat-band zMoSNR the DOS diverge at the onset (where |∇k Ek | = 0) of each subband, also known as vHSs in the DOS Furthermore, the edge states of each subband for 1D crystalline solids can produce a strikingly sharp peak in the DOS spectrum, so that even arbitrary weak interactions at vHS points can yield the strong effects in the optical or transport behavior of electronic devices As shown in Figs 4(a)-4(d), it can be seen that as the barrier height of DBS increases from eV (flat-band) to 0.7 eV, the energy spectrum of DOS evolves from asymmetric vHS peaks to a set of irregularly spaced resonant-tunneling peaks which is a special property of zMoSNRs with DBS These resonant-tunneling peaks of DOS spectrum happen when electron energy E coincides with any resonant-tunneling confined states of DBS, as shown in Figs 4(b)-4(d) A series of fine and discrete confined levels exists in the central channel region, and a continuous distribution of states exists in the left and right contacts Once the coupling between the channel and two contacts occurs, the discrete channel levels obtain part of the contact states which spread into the channel region.18,42 Therefore, the overall coupling effect of two contacts to the channel region is to broaden the DOS in the channel region from its initial discrete energy levels to a continuous DOS structure, as shown in Figs 4(b)-4(d) Furthermore, the broadening of the discrete energy levels inside the channel is proportional to the coupling strength of the contacts (i.e., the broadening term, Γ.).18,42 The contribution to the total DOS of the central channel region originates from the overflowing states of the left and right contacts There are two different DOS contribution to the channel region: one originating from the left contact and weighted by the left electron distribution probability, the other originating from the right contact and weighted by the right electron distribution probability With the increasing of the barrier-height of DBS, the coupling strength of two contacts to the channel region decreases and the line-shape sharpness and depth of the DOS peaks grows, as shown in Figs 4(b)-4(d) Namely, the change of barrier height (i.e., the isolation of the coupling effect) could obviously influence the DOS spectrum in the channel region Therefore, we know that the DBS is a good verified structure for exploring the coupling strength of contacts to channel region which gives rise to the DOS modification of a channel region IV CONCLUSIONS In this paper, for calculating the electronic transport of zMoSNR devices we have developed a full atomic-scale and full quantum method mainly based on NEGF technique in combination with the three-band model We have presented in detail the whole concept and framework of the 085123-12 Chen et al AIP Advances 6, 085123 (2016) NEGF technique We have explored in detail a monolayer MoS2 crystalline solid such as its electron configuration of ion bonds, orbital hybridization, and structure symmetry Therefore, it is reasonable to only adopt three d orbitals as basis for describing a MoS2-based crystalline solid We have explored in detail the three-band model such as its rationality, orbital composition, and microscopic symmetry We have derived in detail the theoretically relational expressions of a zMoSNR crystalline solid and its whole device structure such as their state function, Hamiltonian matrix, and the Schrödinger-like equation in the NEGF form By adopting the complex-band structure method, we have solved the boundary problems of the open boundary system within the NEGF framework Then a powerful method has been developed to solve the quantum transport problems of zMoSNR devices Consequently, the method developed in this paper is expected to be widely adopted due to its conceptual simplicity, computational efficiency, and applied versatility Importantly, the developed method has provided practical formula in detail and in its integrity, which can also be used by experimentalists By using the developed method, we have calculated the physical properties of zMoSNR devices such as their transmission function, conductance, and density of states Furthermore, the important findings in zMoSNR devices have been presented in this calculation such as their conductance quantization, van Hove singularities in the density of states, and contact interaction on channel, which will yield a strong influence on the electronic transport behavior of zMoSNR devices ACKNOWLEDGMENT This work was supported in part by National Science Council (NSC), Taiwan ROC under Contract NSC-99-2112-M-032-006 APPENDIX A: MoS2 MONOLAYER HAMILTONIAN For a MoS2 monolayer, the Hamiltonian, which is calculated using the three-band tight-binding model involving only nearest-neighbor interactions, with orbital ordering {d z 2, d x y , d x 2−y 2} can be written as  h  00 ∗ Hmonolayer(k) =  h01  h∗  02 h01 h11 ∗ h12 h02 h12 , h22 (A1) where 11 11 h00 = 4E11 cos α cos β + 2E11 cos(2α) + ε 1, 22 22 22 + 3E22 ) cos α cos β + 2E11 cos(2α) + ε 2, h11 = (E11 22 22 22 + 3E11 ) cos α cos β + 2E22 cos(2α) + ε 2, h22 = (E22 √ 12 12 12 sin(2α), h01 = −2 3E12 sin α sin β + i2E11 sin α cos β + i2E11 √ 12 12 12 h02 = −2E12 cos α cos β + 2E12 cos(2α) + i2 3E11 cos α sin β, √ 22 22 22 − E11 ) sin α sin β + i4E12 sin α(cos α − cos β), h12 = 3(E22 α = k⊥a, √ β= k ∥ a, k is a wave vector (k⊥, k ∥), a is the lattice constant of MoS2, and a total of eight interaction 11 12 12 22 22 22 parameters exist, namely, ε 1, ε 2, E11 , E11 , E12 , E11 , E12 , and E22 The first two parameters relate to the on-site d-like orbital energies, while the remaining parameters represent the nearest-neighbor interactions 085123-13 Chen et al AIP Advances 6, 085123 (2016) APPENDIX B: ZIGZAG MoS2 NANORIBBON HAMILTONIAN For a zMoSNR with N zigzag lines , the Hamiltonian, which is calculated using the three-band tight-binding model involving only nearest-neighbor interactions, in the |k⊥, j, d > basis can be written in a N × N block matrix form as ( j = 1, 2, · · ·, N)  h1   h2  HzMoSNR(k⊥) =     0 h2+ ··· h1 h2+ h2 h1 ··· h2       ,  h2+  h1  (B1) where h1 and h2 are the × matrices with orbital ordering {d z 2, d x y , d x 2−y 2}, which can be written as 2E 11 cos(2α) + ε 12 12 i2E11 sin(2α) 2E12 cos(2α)   11 12 22 22 sin(2α) 2E11 cos(2α) + ε i2E12 sin(2α)  , h1 =  −i2E11 (B2)  2E 12 cos(2α)  22 22 −i2E12 sin(2α) 2E22 cos(2α) + ε 2 12  √ 12 √ 12 12 11 12   −(E12 ) cos α + 3E11 2E11 cos α i(E11 − 3E12 ) sin α √   √ 12 22 22   22 22 12 22 (E11 + 3E22 ) cos α −i[ (E11 − E22 ) + 2E12 ] sin α  , h2 = −i(E11 + 3E12 ) sin α 2 √   22   −(E 12 − √ 3E 12) cos α −i[ (E 22 − E 22) − 2E 22] sin α 22 (E + 3E ) cos α 11 22 12 11 12   11 22 (B3) and α = 21 k⊥a APPENDIX C: ZIGZAG MoS2 NANORIBBON HAMILTONIAN BETWEEN ADJACENT LAYERS For a zMoSNR with N zigzag lines , the Hamiltonian, which is calculated using the three-band tight-binding model involving only nearest-neighbor interactions, in the |σ, j, d > basis can be written in the N × N block matrix form as ( j = 1, 2, · · ·, N)   q2  Hσ,σ+1 =     0 d 1+    Hσ,σ+2 =     0 g2+ ··· g2+ q2 ··· q2 0 d 1+ d 1+ ··· ··· 0 0      ,  + g2      g2  Hσ,σ−1 =     0      ,    d + d    Hσ,σ−2 =     0 q2+ ··· q2+ g2 0 ··· g2 0 d1 d1 ··· ··· 0 0      ,  + q2    (C1)      ,    d 1 (C2) 085123-14 Chen et al AIP Advances 6, 085123 (2016) and Hσ,σ  z1    =     0 0 z1 z1 ··· 0 ··· 0      ,    z1 (C3) where each of the block matrix elements is × dimensional, in the orbital ordering {d z 2, d x y , d x 2−y 2}, which can be written as 12 √ 12  12 √ 12 (E11 − 3E12 ) − (E12 + 3E11 )   √ 22 22 22 22 22 (E11 + 3E22 ) (E − E11 ) − E12  , 22 √  22 22 22 22 22 (E − E11 ) + E12 (E + 3E11 )  22 22 12 √ 12  12 √ 12 − (E12 − 3E12 ) + 3E11 )  − (E11  √ 22 22 22 22 22 (E11 + 3E22 ) (E − E22 ) + E12  , 11 √  22 22 22 22 22 (E − E22 ) − E12 (E + 3E11 )  11 22  E 11 −E 12 E 12  11 12   11 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