Military Academy, Kaohsiung 830, Taiwan 3Department of Physics, National Cheng Kung University, Tainan 70101, Taiwan 4Center of General Studies, National Kaohsiung Marine University, Kao
Trang 1A 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
Trang 2Quantum transport model for zigzag molybdenum disulfide nanoribbon structures : A full quantum framework
Chun-Nan Chen,1, aFeng-Lin Shyu,2Hsien-Ching Chung,3Chiun-Yan Lin,3
and Jhao-Ying Wu4
1Quantum Engineering Laboratory, Department of Physics, Tamkang University, Tamsui,
New Taipei 25137, Taiwan
2Department of Physics, R.O.C Military Academy, Kaohsiung 830, Taiwan
3Department of Physics, National Cheng Kung University, Tainan 70101, Taiwan
4Center 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 transmis-sion function, conductance, and density of states of zMoSNR devices are calcu-lated 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 Au-thor(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 MX2(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 3The MX2family 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 MX2family 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 (dz2, dx y, and dx2 −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 - dz2, dx y, dx2 −y 2basis given recently by Liu et al.,1is adopted as one of our study founda-tions 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 MoS2monolayer 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.
Trang 3To describe electronic transport through a zMoSNR device, we perform ballistic quantum transport simulations within the non-equilibrium Green’s function (NEGF) formalism15 – 19using a nearest-neighbor three-band model.1The 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 – 22Iterative 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.23–25The 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 combina-tion with the complex energy-band method.26–28The 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 de-vices 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 – 37The 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 1 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= dz 2, dx y, dx2 −y 2) unit-cell-scale basis orbital The three-band matrix elements of mono-layer MoS2Hamiltonian are given as shown inAppendix A.1
FIG 1 Geometric structure of a zMoSNR-based transport device (width : N zigzag lines; channel length: l atomic layers).
Trang 4The state function|k⊥⟩ of a zMoSNR with N zigzag lines (width) in the flat-band region is a linear superposition of the 3 × N terms of tight-binding Bloch basis functions|k⊥, j, d >, which can
be expressed as
|k⊥> =
N
j =1
3
d =1
bj, d(k⊥)|k⊥, j, d >, (1)
where bj, dis the linear expansion coefficient, d denotes the symmetry-type d -like orbital, j spec-ifies 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
bj, d(k⊥, E) is abbreviated for conciseness here Moreover, the tight-binding Bloch basis functions can be written as
|k⊥, j, d > =√1
Ω
σ
exp(ik⊥σa′
)|σ, j, d >, (2) 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 inAppendix B.1Meanwhile, the state function of a zMoSNR can also be written in the form as
|k⊥> =
σ
j, d
where cσ, j,d(k⊥) = √1
Ωbj, 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,σand Hσ,σ′are 3N × 3N matrices (seeAppendix C) whose elements are given by
(Hσ,σ)j, d, j ′ ,d ′= < σ, j, d|(H − E)|σ, j′, d′> (5a) and
(Hσ,σ ′)j, d, j ′ ,d ′= < σ, j, d|H|σ′, j′, d′>, (5b) respectively, and σ′denote σ ± 1 or σ ± 2
The Schrödinger equation (H − E)|k⊥> = 0 in a flat-band zMoSNR can be written in the transfer-matrix form as26 – 28 , 38 – 41
−Hσ,σ−2−1 Hσ,σ−1 −Hσ,σ−2−1 Hσ,σ −Hσ,σ−2−1 Hσ,σ+1 −Hσ,σ−2−1 Hσ,σ+2
cσ−1
cσ
cσ+1
cσ+2
= e−ik ⊥ a′
cσ−1
cσ
cσ+1
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=
cσ,1
cσ,2
cσ,3N
By Bloch’s theorem, the tight-binding coefficients must obey the relation cσ= e−ik ⊥ a ′
cσ+1
Trang 5In the whole structure with perturbation, the wave function of the Schrödinger equation (H
− E)|ψ > = 0 can be expressed as
|ψ > =
k ⊥
a(k⊥)|k⊥> =
σ
j, d
k ⊥
a(k⊥)cσ, j,d(k⊥)|σ, j, d >, (8)
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 > =
σ
j, d
where fσ, j,ddenotes
k ⊥
a(k⊥)cσ, j,d(k⊥), which is electron-energy dependent, i.e., fσ, j,d(E)
Writing the Schrödinger equation (H − E)|ψ > = 0 in the |σ, j, d > basis form, we obtain a linear equation situated at the σth layer as27 , 39 – 41
Hσ,σ−2fσ−2+ Hσ,σ−1fσ−1+ Hσ,σfσ+ Hσ,σ+1fσ+1+ Hσ,σ+2fσ+2= 0, (10) where Hσ,σand Hσ,σ ′are 3N × 3N matrices (seeAppendix C) and fσdenotes a 3N-length column vector whose components are fσ, j,d Therefore, the Hamiltonian matrix of the whole structure can
be written as
H − E ˆ I =
. 0 · · · 0
0 H σ−1,σ−3 H σ−1,σ−2 H σ−1,σ−1 H σ−1,σ H σ−1,σ+1 0 · · · ·
· · · 0 H σ,σ−2 H σ,σ−1 H σ,σ H σ,σ+1 H σ,σ+2 0 · · ·
· · · 0 H σ+1,σ−1 H σ+1,σ H σ+1,σ+1 H σ+1,σ+2 H σ+1,σ+3 0
0 · · · 0
(11)
For a given electron energy E, solving Eq (6) yields a set of 4 ×(3N) real or complex wave vector k⊥, λ; λ= 1,2, ,12N and their associated state functions k⊥, λ which can re-sults 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 , 41while 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 struc-ture 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 > = |ψi> +|ψℜ> = Ii|k⊥,i; L > +
6N
λ=1
a(k⊥ , λ+6N; L)|k⊥ , λ+6N; L > (12a)
and
|ψ; R > = |ψℑ> =
6N
λ=1
a(k⊥, λ; R)|k⊥, λ; R >, (12b)
where Iirepresents 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
Trang 6denote 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
Based on Eq (9), we rewrite it in the form as|ψζ > =
σ
j, d
fσ, j,dζ |σ, j, d >, then we obtain the relation as follows:41
fσζ
fσ+1ζ
=
Bσ,1ζ Bσ,2ζ
Bσ+1,1ζ B
ζ σ+1,2
aζ1
aζ2
where ζ denote ℑ and ℜ for transmitted and reflected waves, respectively,
[Bℑ σ,1 Bσ,2ℑ ] = [cσ,1cσ,2· · · cσ,6N]R, (14a) [Bσ,1ℜ Bσ,2ℜ ] = [cσ,6N +1cσ,6N +2· · · cσ,12N]L, (14b)
aℑ1 =
a(k⊥,1; R)
a(k⊥,2; R)
a(k⊥,3N; R)
and a2ℑ=
a(k⊥ ,3N +1; R)
a(k⊥ ,3N +2; R)
a(k⊥,6N; R)
and
a1ℜ=
a(k⊥ ,6N +1; L)
a(k⊥ ,6N +2; L)
a(k⊥,9N; L)
and aℜ2 =
a(k⊥ ,9N +1; L)
a(k⊥ ,9N +2; L)
a(k⊥,12N; L)
The coefficients fσ, j,dof 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
flℑ+3
flℑ+4
=
Bℑl+3,1 Blℑ+3,2
Bℑl+4,1 Blℑ+4,2
Blℑ+1,1 Blℑ+1,2
Blℑ+2,1 Blℑ+2,2
−1
flℑ+1
flℑ+2
(16a) and
f−3ℜ
f−2ℜ
=
B−3,1ℜ B−3,2ℜ
B−2,1ℜ B−2,2ℜ
Bℜ−1,1 Bℜ−1,2
B0,1ℜ B0,2ℜ
−1
f−1ℜ
f0ℜ
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
Hac t− E ˆI+ ΣL+ ΣR {ϕ}= {S} , (17) where the Hamiltonian of active region (Hact), the boundary self-energies for the left (L) and right (R) regions (ΣL, R), the wave function{ϕ}, and the source term {S} are
Hact− E ˆI =
H0,−1 H0,0 H0,1 H0,2 0 .
H1,−1 H1,0 H1,1 H1,2 H1,3 0 .
0 0
0 Hl +1,l−1 Hl +1,l Hl +1,l+1 Hl +1,l+2
0 · · · 0 0 Hl +2,l Hl +2,l+1 Hl +2,l+2
, (18)
Trang 7
H−1,−3 H−1,−2
0 H0,−2
B−3,1ℜ B−3,2ℜ
B−2,1ℜ B−2,2ℜ
B−1,1ℜ Bℜ−1,2
B0,1ℜ B0,2ℜ
−1
ΣR=
Hl +1,l+3 0
Hl+2,l+3 Hl +2,l+4
Blℑ+3,1 Bℑl +3,2
Blℑ+4,1 Bℑl +4,2
Bℑl+1,1 Blℑ+1,2
Bℑl+2,1 Blℑ+2,2
−1
{ϕ}=
f−1ℜ
f0ℜ
f1
fl
flℑ+1
flℑ+2
and
{S}=
−
0
σ=−3
H−1,σcσ,i
−
0
σ=−2
H0,σcσ,i
−
0
σ=−1
H1,σcσ,i
−H2,0c0, i
0
0
The Green’s function of the device is simply defined as
Gd= E ˆI − Hact− ΣL− ΣR
−1
Within the NEGF formalism: once the Green’s function Gdis found, the transmission function T(E)
is followed by the trace of18 , 42
T(E) = Tr[ΓLGdΓRG+
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(ΣL, 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) = 2e
2
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[ΓLGdΓRG+]|E =E
Trang 8With the Green’s function Gdspecified, the DOS can be obtained via18 , 30 , 42
DOS(E) = 1
2πTr[Gd(ΓL+ ΓR)G+d] (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.1
III RESULTS AND DISCUSSION
The energy-band diagrams of 2D transition-metal dichalcogenides are primarily determined by their crystal structure Structurally, monolayer MoS2can be regarded as a tri-layer S - Mo - S sand-wich, 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 MoS2possess the D3h
(C3, σh, and σv) point-group symmetry
The electron configurations of the Mo and S atoms are Mo :[Kr] 5s14d5and 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 MoS22D hexagonal lattice (D6h), it is important to include the intracell symmetry, especially in the local symmetry inside the trigonal prismatic struc-ture (C3h), which have important effects on the physic properties of monolayer MoS2 Monolayer MoS2structure symmetry indicates that the Mo - 4d and S - 3p orbitals can be classified as three and two groups, respectively, as follows:{dz 2}, {dx 2 −y 2, dx y}, {dx z, dyz} and {px, py}, {pz}.1 , 13 , 14
Moreover, monolayer MoS2D3hsymmetry yields the hybridization of{dz 2} and {dx 2 −y 2, dx 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 consti-tuted by hybridization of the Mo -{dz 2}, {dx 2 −y 2, dx y} orbitals with a little contribution of the S -{px, py} orbitals.1 , 13 , 14Therefore, it is reasonable to only adopt three orbitals (Mo - dz2, dx y, dx2 −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 MoS2which are constituted of not the pure D6hMo- 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 C3hinside 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 MoS2monolayer 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 MoS2can be restored in the three d-orbital model (D6h) by adding E12
11 - and E22
12 -included terms [i.e., the imaginary terms: i2E12
11sin α cos β, i2E12
11sin(2α), i2
√
3E1112cos α sin β, and i4E1222sin α(cos α − cos β)] to the 3 × 3 2D hexagonal Hamiltonian matrix
of the three-band model, as shown inAppendix A In Eq (A1), the E1112- and E1222-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
Trang 9FIG 2 Energy-band diagram for monolayer MoS 2 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 There-fore, the used three-band model is able to capture the essential physics of monolayer MoS2in 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 - dz2, dx y, dx2 −y 2basis, 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 8 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.3 Moreover, the 3N × 3N three-band model
inAppendix Bcan 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 9 of Ref.1).1Note 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 × 4 real (if |e−ik ⊥ a ′
|
= 1) or complex (if |e−ik ⊥ a ′
| , 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 8 zigzag lines,
as shown in the left panel of Fig 3 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∂HN 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.3
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 , 42This term [H − E+ Σ + Σ ] in Eq (17)
Trang 10is not Hermitian, because the boundary self-energies ΣL, R are not Hermitian matrices Conse-quently, 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 ([ΣL+ ΣR] {ϕ} and{S}) Namely, E {ϕ}= Hact{ϕ}+ [ΣL+ ΣR] {ϕ} − {S},18 , 42where[ΣL+ ΣR] {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, Hactdescribes 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− ΣL− ΣR)−1{S} Furthermore, by definition of the NEGF frame-work, a wave function {ϕ} is the Green’s-function response of a source term {S}.18 , 42 Namely, {ϕ}= Gd{S} Consequently, the Green’s function in the active region can be written as Gd
= (E − Hact− ΣL− ΣR)−1, as shown in Eq (22), where the boundary self-energies ΣL (R) describe the interaction of the left (right) contacts to the central channel region of device Therefore, the Green’s function Gd 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 Gd specified, it is easily understood that the transmission function T(E) of the entire device can be expressed in the framework as
T(E) = TrΓLGdΓRG+
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 Gd specified, it is easily understood that the DOS inside the active region of device can be expressed in the framework as DOS(E) = 1
2πTr[Gd(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 = 8 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 Figures4(b)-4(d)compare DOS(E) with T(E) using the
FIG 3 Energy-band diagram for a zigzag-edge MoS 2 nanoribbon with 8 zigzag lines and its corresponding number of propagating channels at a given energy E.