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Tunable spin polarization of MoS2 nanoribbons without time-reversal breaking
Accepted Manuscript Tunable spin polarization of MoS2 nanoribbons without time-reversal breaking Nadia Salami, Aliasghar Shokri
PII:
S0749-6036(17)30964-3
DOI:
10.1016/j.spmi.2017.05.047
Reference:
YSPMI 5028
To appear in:
Superlattices and Microstructures
Received Date: 18 April 2017 Accepted Date: 20 May 2017
Please cite this article as: N. Salami, A. Shokri, Tunable spin polarization of MoS2 nanoribbons without time-reversal breaking, Superlattices and Microstructures (2017), doi: 10.1016/j.spmi.2017.05.047. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Nadia Salami
, Aliasghar Shokri
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Department of Physics, Yasooj Branch, Islamic Azad University, Yasooj, Iran 2
Department of Physics, Payame Noor University, 19395-3697 Tehran, Iran
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May 20, 2017
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Tunable spin polarization of MoS2 nanoribbons without time-reversal breaking
Abstract
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In this work, we present a detailed investigation of the influences of intrinsic spin-orbit coupling (ISOC) as well as Rashba spin-orbit coupling (RSOC) on the spin-resolved transport properties and net spin polarization of molybdenum disulfide (MoS2 ) nanoribbons including different configurations. For this reason, we generalize the tight-binding model by including the effects of the ISOC on all the atoms and a RSOC induced by a vertical electric field. The results predict a noticeable spin polarization in along to the field for the armchair MoS2 nanoribbon close to the Fermi level, which is originated mostly from edge states. Where as, the induced spin polarization in the zigzag MoS2 ribbons is much smaller than that of the considered armchair ones. Contrary to the RSOC, which it can not introduce significant spin polarization in the ZMoS2 with the ribbon width about 1-2 nm, the combination effect of an exchange field and Rashba spin-orbit interaction induces a spin-selective current especially close to the Fermi level. The numerical results may be useful to engineer and design magnetic-field-free spintronic devices based on the MoS2 nanoribbons.
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Keywords: Rashba spin-orbit interaction, Spin polarization, Exchange field, MoS2 nanoribbon, Slater-Koster parameter.
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Corresponding author. E-mail: [email protected], [email protected] E-mail: [email protected]
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Introduction
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The creation and manipulation of the spin current by electrical means is a new route to design the spintronic devices for low-consumption and high-speed operation [1, 2]. There are two usual concepts in the scope of spintronics: intrinsic spin-orbit and Rashba spinorbit interactions. In the prime interaction, it couples the electron’s spin to its motion and provides a spin-polarized current. While, the latter interaction induces a tunable spin-dependent transport. Hence, it is preferred over usual methods such as usage of the ferromagnetic leads or external magnetic fields [3, 4, 5, 6]. There is a strong spin-orbit interaction in the pristine MoS2 monolayer due to Mo heavy element. On the other hand, the lack of inversion symmetry in the pristine MoS2 monolayer causes the valence bands to have been split at K points of Brillouin zone [7]. While, the time-reversal symmetry in the magnetic field-free MoS2 monolayer gives rise to bands with opposite spins at K and −K points. It is favorable for valleytronics and also spin-polarized optical applications [7]. The spin degeneracy of MoS2 monolayer is broken at Γ point due to the Rashba spin-orbit coupling (RSOC) under applying a vertical electric field [8]. Moreover, Lee et al. have predicted a giant Rashba splitting for MoS2 monolayer on Bi(111) bilayer using the density functinal theory (DFT) [9]. The inversion symmetry breaking is physical origin of Rashba spin splitting, which can be provided by creating a non-centrosymmetric potential such as applying a vertical electric field, effect of substrate properties [9, 10], influence of surface states on the surface of bulk [11] and doping a heavy adatom [12]. Moreover, the curvature effect can also induce or enhance a net spin polarization in the presence of RSOC [13]. This coupling can be enhanced by considering some effects such as an exchange field and also electronelectron interaction [14], which the former can be generated due to the proximity with a ferromagnetic insulator substrate [15, 16]. From experimental point of view, MoS2 nanoclusters can be synthesized on the Au(111) possess triangular shape [17]. Also, it is reported that, MoS2 nanoribbons can be fabricated using bottom-up processes, experimentally [18, 19]. This subject has motivated that the researches investigate the properties of MoS2 nanoribbons and their applicants. On the other hand, the graphene nanoribbons exhibit edge-dependent electronic and magnetic properties [20, 21]. These edge-dependent properties can be engineered under an external in-plane transverse electric field or by modified edges via various functional groups [22, 23]. Likewise, it is reported that, the electronic, magnetic and transport properties of MoS2 nanoribbons are strongly dependent on the special geometry of their edges [24, 25]. In our recent work, we studied the spin-less electronic properties of two types of MoS2 nanoribbons in the presence of vacancies and also weak scatters [25]. Also, we have introduced the tunable electronic and magnetic properties of a MoS2 monolayer with two types of intrinsic and extrinsic vacancies under elastic planar strain at the DFT level of theory [26]. Beside these, in the earlier work we have computed the electronic transport properties of the MoS2 monolayer transducer upon exposure to CO, CO2 and NO gas adsorbates using the non-equilibrium Greens function formalism (NEGF) based on DFT [27]. According to a recent report, the zigzag MoS2 nanoribbon exhibits an half-metal behavior in exposure to a combination of an external exchange field and a transverse external electric field based on multi-band tight binding approximation [28]. So, magnetic-field-free spintronic devices can be expected and proposed based on two types of defect-free MoS2 nanoribbons due to their edge topology. 2
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2.1
Description of model and theoretical framework
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In this paper, we address that in response to a vertical electric field, the edge states in the MoS2 nanoribbons can induce spin-resolved currents. These spin currents can be controlled by means of a vertical electric field. Besides, to reveal the capabilities of MoS2 nanoribbons as a spin polarizer, we numerically investigate the ISOC as well as the RSOC effects on the spin polarization of two types of MoS2 nanoribbons. In details, our numerical calculations are performed using the NEGF within multi-band tight-binding method. In this regard, the total density of states (TDOS), the band structure, spin-resolved transmission spectrums and spin polarization of MoS2 nanoribbons are investigated in the presence of the electric field induced Rashba spin-orbit interaction. Also, in order to achieve an optimum spin polarization in the studied nanostructures, the effect of a uniform exchange field is investigated in the presence of the RSOC spin-orbit coupling. The paper is organized as follows. In Section 2, an applied computational model is described to efficiently calculate spin transport in MoS2 nanoribbons in the presence of Rashba spin-orbit interaction and also intrinsic spin-orbit interaction. In Section 3, the electronic and transport properties of two types of MoS2 nanoribbons are studied with and without ISOC and also the RSOC. The spin polarization of the systems are investigated with different edge terminations and symmetries. Also the band gap evolution of the systems are studied due to the different values of the Rashba coefficient. Finally, by a combination of a Rashba spin-orbit interaction and an exchange field effects it is attempted to predict controllable and noticeable spin polarized systems. The last section is briefly devoted to a discussion and conclusion of our findings.
Tight-binding description of MoS2 monolayer
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In order to propose spin-polarized system, we introduce a planar two-terminal device based on MoS2 nanoribbons as shown in Figs. 1(a) and 2(a). The quasi one-dimensional device is composed of semi-infinite left and right electrodes and also central region, which three regions include MoS2 nanoribbon with same widths. Whereas, spin-orbit coupling exists only in the central region and the conductance of the central region will be studied in the presence of Rashba spin-orbit interaction induced by applying an external electric field perpendicular to the ribbon. The electronic properties of MoS2 monolayer close to the Fermi level (at K and K 0 points) or even the whole Brillouin zone can be described using various proposed multiband Slater-Koster tight binding schemes which derived from first-principles studies [7, 29, 30, 31, 32, 33, 34]. It is reported that the low energy Bloch states of MoS2 monolayer are mostly made of d and p orbitals for Mo and S, respectively. However, Mo orbitals possess dominant components in both conduction and valence bands. Even though, the existence of orbitals due to the S atoms in the top and bottom plane causes the inversion symmetry to break in the MoS2 monolayer. So, in order to model the system with proper symmetry, the Hamiltonian can be described in atomic orbital basis sets including dz2 , dx2 −y2 , dxy , dyz and dxz for Mo and px , py and pz for S atoms. So, the Hamiltonian of the system in real space can be presented as XX HSK = [(εnµ δnm δµν + tnµ,mν )a†nµ amν + h.c.]. (1) n,m µ,ν
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Here, a†nµ (amν ) indicates the creation (annihilation) operator of µ(ν)-th atomic orbital at n(m)-th atom. Also, εnµ and tnµ,mν describe elements of on-site matrices of S and Mo atoms and elements of hopping matrices between first-nearest neighbor atoms, respectively. Using the Slater-Koster transformations, these elements of hopping matrices are derived from direction cosines and the related Slater-Koster tight-binding parameters [35]. The related parameters of pdσ and pdπ for the nearest-neighbors of Mo-S, ddσ, ddπ and ddδ for nearest-neighbor of Mo-Mo, and also ppπ and ppσ for nearest-neighbor of S-S are given in Table 1. The armchair and zigzag MoS2 nanoribbons can be specified by number of dimer lines Na and number of zigzag lines Nz across nanoribbon, respectively. An ideal MoS2 nanoribbon is constructed by repeating periodically unit-cell of nanoribbon only along one direction. Additionally, by taking account only interaction with first nearest-neighbor unitcells, the Hamiltonian in k space will be obtained as H(kx ) = Hl−1,l exp(−ikx Tx ) + Hl,l + Hl,l+1 exp(ikx Tx ).
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Here, Hl,l and Hl,l+1 describe the Hamiltonian of interactions between atoms of l-th unitcell and between atoms of l-th with atoms of l + 1-th unit-cell, respectively. Also, Tx indicates lattice constant of the nanoribbon. It should be noted that, the band structure is obtained from eigenvalue calculations of above matrix. Furthermore, Hl,l and Hl,l+1 matrices are 11N ×11N blocks, which N is Na (Nz ) in the armchair (zigzag) nanoribbon.
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The intrinsic Spin-orbit interaction
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In the presence of the ISOC, the Hamiltonian of the system can be given in the spin space by H = HSK ⊗ 12 + HISOC ,
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∆α Lα · Sα . h ¯
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HαISOC =
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where, as afore-mentioned in earlier subsection, HSK is the Hamiltonian system in the absence of the spin-orbit interaction. The second term represents the Hamiltonian of the intrinsic spin-orbit interaction, which for each site is given by
Here, ∆α is the ISOC constant, which depends on the type of specific atom (α = Mo and S). Also, L and S are orbital and spin angular momentum operators, respectively. Here, the spin-orbit interaction effect can be evaluated using the representation − − + ∆α L+ α Sα + Lα Sα α z z (5) HISOC = + Lα Sα h ¯ 2 where, +
S =h ¯
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h ¯ , S = 2 z
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The orbital angular momentum operator L acts on |l, mi state as p L± |l, mi = h ¯ l(l + 1) − m(m ± 1)|l, m + 1i, Lz |l, mi = m¯h|l, mi. 4
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Here, l and m denote the quantum number of angular momentum and its z component, respectively. The chosen basis sets can be presented as 1 i |pz i = |1, 0i, |px i = − √ (|1, 1i − |1, −1i), |py i = √ (|1, 1i + |1, −1i), 2 2
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and
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|d3z2 −r2 i = |2, 0i , |dx2 −y2 i = √12 (|2, 2i + |2, −2i), |dxy i = − √i2 (|2, 2i − |2, −2i), |dxz i = − √12 (|2, 1i − |2, −1i), |dyz i = √i2 (|2, 1i + |2, −1i).
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The ISOC term is taken into account quantum effects of interaction of spin with orbital motion of electron. The hopping matrices of the HISOC block (matrix elements of the spinorbit coupling Hamiltonian) between different atoms are assumed to be zero, because the spin-orbit interaction has its largest effect on carriers. So, the intra-atomic interaction is acted on all atoms which its value depends on the atom specie. The on-site matrices of the HISOC block act on the basis states |d3z2 −r2 i, |dx2 −y2 i, |dxy i, |dxz i and |dyz i √ √ 3 3 0 0 0 − i 0 0 0 0 0 2 2 0 1 1 0 0 −i 0 0 0 0 i 2 2 1 1 Mo Mo 0 0 −2i εISOC,↑↑ = 0 i 0 0 0 , εISOC,↑↓ = √0 2 , 0 0 0 0 −i − 12 21 i 0 0 √23 2 − 3 0 0 0 2i 0 i − 12 i − 12 0 0 2 (10) √ √ 3 3 0 0 0 i 0 0 0 0 0 2 2 0 1 1 0 0 i 0 0 i 0 0 − 2 2 0 1 1 Mo Mo 0 0 − i − εISOC,↓↓ = 0 −i 0 0 0 , εISOC,↓↑ = √ 2 2 . − 3 1 1 0 0 0 0 i i 0 0 2 2 2 2 √ i 3 − 1 1 0 0 0 −2 0 i −2i 2 0 0 2
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In a similar way, the resulting on-site matrices states |px i, |py i and |pz i as 0 − 21 i 0 0 S S 1 0 εISOC,↑↑ = 2 i 0 0 , εISOC,↑↓ = 0 0 0 − 12 1 0 i 0 0 2 S S 1 εISOC,↓↓ = − 2 i 0 0 , εISOC,↓↑ = 0 1 0 0 0 2
of the block HISOC act on the basis 1 0 2 0 − 12 i , 1 i 0 2 0 − 12 0 − 12 i . 1 i 0 2
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The Hamiltonian of the intrinsic spin-orbit interaction is given in the spin space by ↑↑ HISOC H↑↓ ISOC HISOC = . (12) ↓↓ H↓↑ ISOC HISOC So, the intra-atomic ISOC Hamiltonian becomes a 22N ×22N matrix, which N is Na (Nz ) in the armchair (zigzag) nanoribbon. 5
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2.3
The Rashba Spin-orbit interaction
Under applying an electric field, the Hamiltonian of the Rashba spin-orbit interaction in a continuum space is given by E ) ≡ tR z · (σ×k), (13) 2mc2 where, µB and σ are the Bohr magneton and spin Pauli matrices, respectively. In addition, the Rashba coefficient tR , indicates the strength of RSOC, which evidently is proportional to the gradient potential. Corresponding tight-binding Hamiltonian gets the form XX HRSOC = (itR c†n,µ [σ × dˆn,n+1 ]cn+1,ν ). (14) n
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HRSOC = −µB σ · (p ×
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Here, dˆn,n+1 describes the unit vector connecting of site n to site n + 1. The Hamiltonian of the Rashba spin-orbit interaction mixes different spins and gives rise to flip the spin of electron from up to down, which in the spin space takes the following form 0 H↑↓ RSOC HRSOC = . (15) H↓↑ 0 RSOC It should be added that, the inter-atomic RSOC Hamiltonian, 22N ×22N matrix, can be split into the hopping matrices acting on the basis states corresponding to first nearestneighbor atoms. The Rashba hopping integrals matrices are measured in the unit of corresponding Slater-Koster hopping matrices as (16)
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↓↑ t↑↓ nm,RSOC = βtnm , tnm,RSOC = γtnm ,
where β =p0.5itR cos(φ)[sin(θ)+i cos(θ)], and γ = 0.5itR cos(φ)[sin(θ)−i cos(θ)]. In which cos(φ) = 4/7 and also θ is the angle between in-plane nearest-neighbor atoms in the MoS2 monolayer [30].
Spin-polarized current
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Here, we investigate the electrical transport of the planar devices in the presence of Rashba spin-orbit interaction, with a combination of spatial symmetrical relations and time-reversal symmetry are considered. It was been explained in Ref. [36] and we discuss about these symmetrical relations at next section. However, the numerical studies can reveal strength of spin polarization of different systems under study. The device is assumed including central region in the presence of both types of spinorbit interactions and two semi-infinite leads in the absence of any type of spin-orbit coupling. The transmission of the device is computed within the NEGF method [37, 38]. The transmission probability of an incident electron through a device is obtained as a function of incident energy from the extended Green’s function of the central region, GC , as T (E) = Tr[ΓL G†C ΓR GC ].
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The Greens function of the central region coupled to the left and right electrodes is given by GC (E) = [(E + i)I − HC − ΣL (E) − ΣR (E)]−1 , 6
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where, is an arbitrarily small number added into the above equation to induce the convergence criterion and I is the identity matrix. Here, the broadening function ΓL(R) indicates the coupling between the device and the left (right) electrode. ΓL(R) = i[Σ†L(R) − ΣL(R) ].
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In above consideration, ΣL(R) is the self-energy of the left (right) electrode, and it is calculated from the surface Green’s function of the left (right) electrode and interaction between the central region and the left (right) electrode gL(R) via ΣL (E) = HLC † gL HLC , ΣR (E) = HCR gR H†CR .
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In order to calculate the surface retarded Green’s function of the semi-infinite left and right leads, we use the transfer matrix technique as [25, 39, 40]. −1
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(g0,0 )L = [(E + i)I − H0,0 − H−1,0 † Λ] ,
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Here, Λ and Λ are the transfer matrices, which can be efficiently calculated within an iterative scheme as defined by Refs. [25, 39, 40]. In Eq. (21), H0,0 represents the Hamiltonian of the surface unit-cell and H0,1 is its interaction with the adjacent supercells. With above consideration, we can calculate the TDOS of the system at the energy E as 1 TDOS(E) = − ImTr[GC (E)]. π
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Moreover, in the spin space, the spin-resolved transmission coefficient is given by 0
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T σσ = T r(Tσσ ), 0
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σ σσ † σσ where Tσσ = Γσ0 Gσσ denotes transmission probability of electron with 11 Γ2 (G11 ) . T 0 spin σ through the central region when the electron with spin σ is injected into the region, in the presence of both spin-flip and spin-conserved processes. It can be rewritten in the spin space as
T↑↑ T↑↓ T↓↑ T↓↓
↑↑ ↑ ↑ −1 ΣR 0 H H↑↓ ΣL 0 − . (24) = (E + i)I − − H↓↑ H↓↓ 0 Σ↓R 0 Σ↓L
Computationally, the transmission T↑↑ and T↑↓ can be obtained, when one considers that, only spin-up electrons inject from the left electrode into the central region. Finally, the spin polarization along to z direction, (SP )z , is defined as (SP )z =
(T ↑↑ + T ↓↑ − T ↑↓ − T ↓↓ ) . (T ↑↑ + T ↓↑ + T ↑↓ + T ↓↓ )
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Calculated results and discussions
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The main purpose of the work is to investigate theoretically electronic properties of the AMOS2 and ZMOS2 nanoribbons in the absence and in the presence of the ISOC and also the RSOC effects. The whole of energy parameters has been scaled dimensionless. Also, c, e and h are set to 1. Besides, an in-plane interatomic spacing is taken 0.316 nm according to the experimental evidence [30]. Therefore, the lattice constant Tx of the AMoS2 and ZMOS2 nanoribbons are selected as 0.547 nm and 0.316 nm, respectively. The band structure of nanoribbons is calculated numerically using 50 k-points sampling along to Γ − X direction. Additionally, using Eqs. (18)-(22), the TDOS of the nanoribbons will be obtained in the presence of spin-orbit coupling, when all three regions (two electrodes and the central region) are affected due to them. Also, in all calculations, the length of central region is assumed to be one unit-cell. Since, the considered length for the central region is much smaller than 14-18 nm, which this length is the mean free path of MoS2 monolayer [41, 42]. So, the transport properties of the system are calculated in the ballistic regime in the absence of any scattering process. In addition, the arbitrarily small number, is taken 0.0002 eV. The ISOC constant for Mo (∆Mo ) and S (∆S ) atoms is taken 0.075 eV and 0.052 eV, respectively [31]. Let us start by investigating the influence of ISOC as well as electric field induced RSOC on the electronic properties of the MoS2 nanoribbons. The number of the dimer line of AMoS2 ribbon and the zigzag line of ZMoS2 ribbon is given by Na = 10 and Nz = 6, respectively as schematized in Figs. 1(a) and 2(a). These two nanoribbons possess nearly same width 1.420 nm and 1.457 nm, respectively. Figs. 1 and 2 show the band structure and TDOS of two types of MoS2 nanoribbons in the absence and also in the presence of two types of spin-orbit couplings. Generally, the figures clearly show that the electronic properties of MoS2 nanoribbons strongly depend on the type of nanoribbon with different edge terminations. Figs. 1(b) and 2(b) reveal that, both of the armchair and zigzag nanoribbons under study exhibit n−type semiconductor behavior, which possess the bandgap of about 1.56 eV and 1.04 eV, respectively. The spin-less electronic properties of the nanoribbons are discussed completely at greater length in Ref. [25]. Moreover, the dangling bonds of edge atoms induce some mid-gap states close to the conduction band in the AMoS2 ribbon, which these mid-gap states localize mainly on S atoms. While, two Mo edge atoms have dominant association to constitute top of the valence band in the AMoS2 nanoribbon (the figure is not shown). The origin of these can be related to the chemical nature of S and Mo atoms. Indeed, S atom is more electronegative with respect to Mo atom. So, this causes electrons to accumulate on S atoms at the edge which, acts as adatoms. Contrary to the armchair nanoribbon, the states due to only S (Mo) atoms at the edge constitute the bottom (top) of conduction (valence) band in the ZMoS2 nanoribbon, which this atomic contribution is unaffected due to the RSOC. In addition, Figs. 1(c) and 1(d) (2(c) and 2(d)) in comparison to Fig. 1(b) (2(b)) illustrate that, both of intrinsic and Rashba spin-orbit couplings lift band degeneracy and split bands of the AMoS2 (ZMoS2 ) nanoribbon into subbands. It is seen that the band spin splitting enhances when both of them are assumed. The reason of this splitting is explained as follows: existence of both time-reversal and spatial inversion symmetries in the system simultaneously gives rise to spin degeneracy of the band structure [11]. For example, an fcc lattice has an inversion symmetry. Therefore, the band degeneracy of this lattice (e.g., Au) is preserved in the presence of spin-orbit interaction. While, the spin 8
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degeneracy of bands is removed, when there is spin-dependent term in the Hamiltonian of a system without inversion symmetry. On the other hand, the time-reversal symmetry is preserved in the absence of an external magnetic field or magnetic impurities [36]. Also, the spin-orbit interactions lead to broadening of the density of states to some extent. Besides, Fig. 1(d) (2(d)) suggests that the RSOC with the Rashba coefficient tR = 0.1 does not cause the band gap to change noticeably in the AMoS2 (ZMoS2 ) ribbon. Moreover, the effects of Rashba coefficient intensity are studied on the electronic properties of MoS2 nanoribbons in Fig. 3. In this regard, the band structure and the TDOS of the AMoS2 and ZMoS2 nanoribbons are calculated for two different values of the Rashba coefficient, 0.1 and 0.5. The figure clearly demonstrates that the band spin splitting and also broadening of the TDOS enhance when the intensity value of Rashba coefficient increases. Evidently, the reason of this enhancement is related to the increase of spin-flip rate due to higher intensity value, which will be discussed more at next paragraph. Fig. 4 displays the spin-resolved transmission, spin polarization (SP )z and band structure of AMoS2 nanoribbon as a function of carrier energy in the presence of combination of the Rashba and also intrinsic spin-orbit couplings. Here, the numerical calculations are performed with tR = 0.5. By referring to the Fig. 4(a), we can see that the T ↑↓ and T ↓↑ appear noticeably in the presence of the RSOC. Indeed, the effect causes spin-flipping process to occur for the transmitted carriers through the channel. One should note that, T↑↓ and T↑↑ indicate spin-flipping and spin-conserving processes during carrier transport through Rashba device, respectively. According to Figs. 4(b) and 4(c), both of the incident electrons at the lowest subbands and the incident holes at the highest subbands around the Fermi level contribute to negative spin polarization (SP )z . So, by confining the electrons to occupy only these subbands, the spin-polarized current is generated and the system can be used as a spin filter device. However, the predicted spin-polarized current occurs due to the electric field induced RSOC in other quasi-one dimensional systems such as graphene nanoribbons and carbon nanotubes [36, 43]. Besides, the spin polarization of AMoS2 ribbon transmission close to the Fermi level is originated mostly from the edge and mid-gap states. Moreover, as shown in the panels (c)-(e) of Fig. 4, the spin splitting of the bands increases significantly, when the RSOC is combined to the inherent coupling. In order to gain a better understanding of the effect of the RSOC on the transport properties of zigzag ribbons, we have calculated the spin-dependent transmission, spin polarization (SP )z and band structure of ZMoS2 ribbon as a function of the incident electron energy in Fig. 5. From comparison Figs. 4a to 5a, we can see that the spinflipping process for the ZMoS2 ribbon occurs less than the AMoS2 ribbon particularly close to the Fermi level. Intriguingly, the RSOC does not introduce a noticeable spin polarization in the ZMoS2 ribbon even close to the Fermi level. While, the lowest subband of the incident electrons in the ZMoS2 ribbon splits significantly, when the RSOC with tR = 0.5 is combined to intrinsic one (see Fig. 5(c)). The reason may be related to construction of the lowest and highest subbands, which originates from only Mo and S edge atoms. Therefore, one can conclude that the S and Mo edge atoms do not have any effect to generate a spin-polarized system based on the zigzag nanoribbon under study in the exposure to the electric-field induced RSOC. In order to explore the influences of edge termination of the Rashba devices based on two types of nanoribbons, we investigate the spin polarization of several different widths and edge terminations: Na = 8, Na = 10 and Na = 12 for AMoS2 , and Nz = 5, Nz = 6 9
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and Nz = 7 for ZMoS2 ribbons. Fig. 6 clearly shows that the induced spin polarization of ZMoS2 nanoribbons is much smaller than that of the considered armchair ones. Besides, the position of energy peaks depends to large extent on the type of ribbon. Also, the most important difference between spin polarization of different armchair nanoribbons close to the Fermi level is that the contribution of incident electrons at the lowest conduction subbands changes from negative spin polarization to positive spin polarization, when the dimmer lines varies from Na =10 to Na =12. Fig. 7 shows the band gap evolution of two types of MoS2 ribbons due to the RSOC with different Rashba coefficient. The RSOC induces gradual decrease in the band gap of the AMoS2 and ZMoS2 ribbons when the Rashba coefficient increases. Whereas, the slop of variation is nearly same for both nanoribbons under study. Finally, we examine the effect of a uniform exchange field on the spin polarization of MoS2 ribbons in the presence of the RSOC. The exchange field may be originated from the proximity with a ferromagnetic insulator substrate or from depositing ferromagnetic atoms at the edge of ribbon. Hence, this breaks the time-reversal symmetry. The Hamiltonian of the Rashba device [28] in the central region becomes X a†nµ σz anµ , (26) H0 = H − M n,µ
Concluding remarks
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when the exchange field is taken into account. In which, M indicates the strength of the exchange field. In Fig. 8, we depicted the spin polarization of AMoS2 and ZMoS2 nanoribbons in exposure to combination effects of the intrinsic and the RSOC and also a uniform exchange field. The numerical calculations are carried out for tR = 0.5 and M = 2.4∆α . Generally, the exchange filed causes the spin polarization to increase in the studied nanoribbons to some extent, which this increase is more pronounced close to the Fermi level. Despite to the RSOC, which it can not introduce the noticeable spin polarization in the ZMoS2 , combination of the exchange field and the RSOC induce spinselective current especially close to the Fermi level. Such that, the spin polarization of ZMoS2 reaches 100% via considering the exchange field close to the Fermi level.
Our main focus in this work was the role of the electric field induced Rashba spin-orbit interaction in introducing spin-resolved transport in the planar devices based on MoS2 nanoribbons. The results show that, regardless of significant spin splitting in the lowest conduction bands of the zigzag MoS2 nanoribbons, the Rashba-type spin-orbit interaction cannot induce significant spin polarization in the output electrode of a two-terminal system based on the planar zigzag MoS2 ribbons with about width of 1.457 nm and different edge terminations. While a net spin polarization in the armchair MoS2 ribbons with nearly same width is provided mostly close to the Fermi level which, the predicted net spin polarization is originated mostly from edge states. Whereas, the induced spin polarization enhances, when applying a vertical electric field is combined to proximity of a ferromagnetic insulator substrate. Such that, the spin polarization of ZMoS2 (6) reaches 100% close to the Fermi level via considering the exchange field. Therefore, the tunable spin-dependent transport can be provided in the MoS2 nanoribbons.
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Acknowledgment
This work has been supported by Yasooj Branch, Islamic Azad University through a research fund.
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Figure 1: (a) Schematic representation of AMoS2 nanoribbon with width of 1.420 nm, the band structure and the TDOS in (b) the absence and in the presence of (c) the ISOC and also (d) the RSOC with the tR = 0.1. The unit-cell of the nanoribbon is composed of 30 atoms and is identified by the black line. The red (gray) circle denotes the S (Mo) atom.
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Figure 2: (a) Schematic representation of ZMoS2 nanoribbon with width of 1.457 nm, the band structure and the TDOS in (b) the absence and in the presence of (c) the ISOC and also (d) the RSOC with the tR = 0.1. The unit-cell of the nanoribbon is composed of 18 atoms and is identified by the black line. The red (gray) circle denotes the S (Mo) atom.
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Figure 3: The dependence of the band structure, TDOS as a function of the Rashba coefficient for (a) AMoS2 nanoribbon with Na = 10 and tR =0.1 and (b) tR =0.5 and also for (c) ZMoS2 nanoribbon with Nz = 6 and tR =0.1 and (d) tR =0.5. Figure 4: (a) Spin-resolved transmissions, (b) spin polarization with respect to the z direction, (SP )z and (c) band structure of AMoS2 nanoribbon with Na = 10 under a combined effect of the RSOC and the ISOC with tR = 0.5. Also band structure of AMoS2 nanoribbon (d) in the presence and (e) in the absence of the ISOC is presented.
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Figure 5: (a) Spin-resolved transmissions, (b) spin polarization with respect to the z direction, (SP )z and (c) band structure of ZMoS2 nanoribbon with Nz = 6 under a combined effect of the RSOC and the ISOC with tR = 0.5. Also band structure of ZMoS2 nanoribbon (d) in the presence and (e) in the absence of the ISOC is presented.
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Figure 6: Spin polarization, (SP )z , as a function of energy for AMoS2 nanoribbon with (a) Na = 8, (b) Na = 10, (c) Na = 12 and for ZMoS2 nanoribbon with (d) Nz = 5, (e) Nz = 6 and (f) Nz = 7 in the presence of the RSOC and the ISOC. In all cases, the Rashba coefficient is tR = 0.5. Figure 7: The dependence of the band gap of AMoS2 (Na = 10) and ZMoS2 (Nz = 6) on the Rashba coefficient. Figure 8: Spin polarization of (a) AMoS2 (Na = 10), (b) ZMoS2 (Nz = 6) nanoribbons in the presence of the RSOC and also the exchange field for M = 2.4∆α . The Rashba coefficient of tR = 0.5 is taken. Table 1: εMo 1 -1.512 ppσ 0.696
The Slater-Koster parameters for MoS2 monolayer are given in unit of eV [31]. εMo εMo εMo εMo εS1 εS2 εS3 2 3 4 5 -3.025 -3.025 0.419 0.419 -1.276 -1.276 -8.236 ppπ ddσ ddπ ddδ pdσ pdπ 0.278 -0.933 -0.478 -0.442 -2.619 -1.396 13
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• Two types of defect-free MoS2 nanoribbons have been investigated. • The vertical electric field and edge states in the system induce spin-resolved currents. • The creation and manipulation of the spin current by electrical means is a route to design the spintronics devices for low-consumption and high-speed operation. • Our calculations are based on NEGF within multi-band tight-binding method.
PII:
S0749-6036(17)30964-3
DOI:
10.1016/j.spmi.2017.05.047
Reference:
YSPMI 5028
To appear in:
Superlattices and Microstructures
Received Date: 18 April 2017 Accepted Date: 20 May 2017
Please cite this article as: N. Salami, A. Shokri, Tunable spin polarization of MoS2 nanoribbons without time-reversal breaking, Superlattices and Microstructures (2017), doi: 10.1016/j.spmi.2017.05.047. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Nadia Salami
, Aliasghar Shokri
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Department of Physics, Yasooj Branch, Islamic Azad University, Yasooj, Iran 2
Department of Physics, Payame Noor University, 19395-3697 Tehran, Iran
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May 20, 2017
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Tunable spin polarization of MoS2 nanoribbons without time-reversal breaking
Abstract
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In this work, we present a detailed investigation of the influences of intrinsic spin-orbit coupling (ISOC) as well as Rashba spin-orbit coupling (RSOC) on the spin-resolved transport properties and net spin polarization of molybdenum disulfide (MoS2 ) nanoribbons including different configurations. For this reason, we generalize the tight-binding model by including the effects of the ISOC on all the atoms and a RSOC induced by a vertical electric field. The results predict a noticeable spin polarization in along to the field for the armchair MoS2 nanoribbon close to the Fermi level, which is originated mostly from edge states. Where as, the induced spin polarization in the zigzag MoS2 ribbons is much smaller than that of the considered armchair ones. Contrary to the RSOC, which it can not introduce significant spin polarization in the ZMoS2 with the ribbon width about 1-2 nm, the combination effect of an exchange field and Rashba spin-orbit interaction induces a spin-selective current especially close to the Fermi level. The numerical results may be useful to engineer and design magnetic-field-free spintronic devices based on the MoS2 nanoribbons.
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Keywords: Rashba spin-orbit interaction, Spin polarization, Exchange field, MoS2 nanoribbon, Slater-Koster parameter.
∗ †
Corresponding author. E-mail: [email protected], [email protected] E-mail: [email protected]
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Introduction
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The creation and manipulation of the spin current by electrical means is a new route to design the spintronic devices for low-consumption and high-speed operation [1, 2]. There are two usual concepts in the scope of spintronics: intrinsic spin-orbit and Rashba spinorbit interactions. In the prime interaction, it couples the electron’s spin to its motion and provides a spin-polarized current. While, the latter interaction induces a tunable spin-dependent transport. Hence, it is preferred over usual methods such as usage of the ferromagnetic leads or external magnetic fields [3, 4, 5, 6]. There is a strong spin-orbit interaction in the pristine MoS2 monolayer due to Mo heavy element. On the other hand, the lack of inversion symmetry in the pristine MoS2 monolayer causes the valence bands to have been split at K points of Brillouin zone [7]. While, the time-reversal symmetry in the magnetic field-free MoS2 monolayer gives rise to bands with opposite spins at K and −K points. It is favorable for valleytronics and also spin-polarized optical applications [7]. The spin degeneracy of MoS2 monolayer is broken at Γ point due to the Rashba spin-orbit coupling (RSOC) under applying a vertical electric field [8]. Moreover, Lee et al. have predicted a giant Rashba splitting for MoS2 monolayer on Bi(111) bilayer using the density functinal theory (DFT) [9]. The inversion symmetry breaking is physical origin of Rashba spin splitting, which can be provided by creating a non-centrosymmetric potential such as applying a vertical electric field, effect of substrate properties [9, 10], influence of surface states on the surface of bulk [11] and doping a heavy adatom [12]. Moreover, the curvature effect can also induce or enhance a net spin polarization in the presence of RSOC [13]. This coupling can be enhanced by considering some effects such as an exchange field and also electronelectron interaction [14], which the former can be generated due to the proximity with a ferromagnetic insulator substrate [15, 16]. From experimental point of view, MoS2 nanoclusters can be synthesized on the Au(111) possess triangular shape [17]. Also, it is reported that, MoS2 nanoribbons can be fabricated using bottom-up processes, experimentally [18, 19]. This subject has motivated that the researches investigate the properties of MoS2 nanoribbons and their applicants. On the other hand, the graphene nanoribbons exhibit edge-dependent electronic and magnetic properties [20, 21]. These edge-dependent properties can be engineered under an external in-plane transverse electric field or by modified edges via various functional groups [22, 23]. Likewise, it is reported that, the electronic, magnetic and transport properties of MoS2 nanoribbons are strongly dependent on the special geometry of their edges [24, 25]. In our recent work, we studied the spin-less electronic properties of two types of MoS2 nanoribbons in the presence of vacancies and also weak scatters [25]. Also, we have introduced the tunable electronic and magnetic properties of a MoS2 monolayer with two types of intrinsic and extrinsic vacancies under elastic planar strain at the DFT level of theory [26]. Beside these, in the earlier work we have computed the electronic transport properties of the MoS2 monolayer transducer upon exposure to CO, CO2 and NO gas adsorbates using the non-equilibrium Greens function formalism (NEGF) based on DFT [27]. According to a recent report, the zigzag MoS2 nanoribbon exhibits an half-metal behavior in exposure to a combination of an external exchange field and a transverse external electric field based on multi-band tight binding approximation [28]. So, magnetic-field-free spintronic devices can be expected and proposed based on two types of defect-free MoS2 nanoribbons due to their edge topology. 2
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2.1
Description of model and theoretical framework
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In this paper, we address that in response to a vertical electric field, the edge states in the MoS2 nanoribbons can induce spin-resolved currents. These spin currents can be controlled by means of a vertical electric field. Besides, to reveal the capabilities of MoS2 nanoribbons as a spin polarizer, we numerically investigate the ISOC as well as the RSOC effects on the spin polarization of two types of MoS2 nanoribbons. In details, our numerical calculations are performed using the NEGF within multi-band tight-binding method. In this regard, the total density of states (TDOS), the band structure, spin-resolved transmission spectrums and spin polarization of MoS2 nanoribbons are investigated in the presence of the electric field induced Rashba spin-orbit interaction. Also, in order to achieve an optimum spin polarization in the studied nanostructures, the effect of a uniform exchange field is investigated in the presence of the RSOC spin-orbit coupling. The paper is organized as follows. In Section 2, an applied computational model is described to efficiently calculate spin transport in MoS2 nanoribbons in the presence of Rashba spin-orbit interaction and also intrinsic spin-orbit interaction. In Section 3, the electronic and transport properties of two types of MoS2 nanoribbons are studied with and without ISOC and also the RSOC. The spin polarization of the systems are investigated with different edge terminations and symmetries. Also the band gap evolution of the systems are studied due to the different values of the Rashba coefficient. Finally, by a combination of a Rashba spin-orbit interaction and an exchange field effects it is attempted to predict controllable and noticeable spin polarized systems. The last section is briefly devoted to a discussion and conclusion of our findings.
Tight-binding description of MoS2 monolayer
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In order to propose spin-polarized system, we introduce a planar two-terminal device based on MoS2 nanoribbons as shown in Figs. 1(a) and 2(a). The quasi one-dimensional device is composed of semi-infinite left and right electrodes and also central region, which three regions include MoS2 nanoribbon with same widths. Whereas, spin-orbit coupling exists only in the central region and the conductance of the central region will be studied in the presence of Rashba spin-orbit interaction induced by applying an external electric field perpendicular to the ribbon. The electronic properties of MoS2 monolayer close to the Fermi level (at K and K 0 points) or even the whole Brillouin zone can be described using various proposed multiband Slater-Koster tight binding schemes which derived from first-principles studies [7, 29, 30, 31, 32, 33, 34]. It is reported that the low energy Bloch states of MoS2 monolayer are mostly made of d and p orbitals for Mo and S, respectively. However, Mo orbitals possess dominant components in both conduction and valence bands. Even though, the existence of orbitals due to the S atoms in the top and bottom plane causes the inversion symmetry to break in the MoS2 monolayer. So, in order to model the system with proper symmetry, the Hamiltonian can be described in atomic orbital basis sets including dz2 , dx2 −y2 , dxy , dyz and dxz for Mo and px , py and pz for S atoms. So, the Hamiltonian of the system in real space can be presented as XX HSK = [(εnµ δnm δµν + tnµ,mν )a†nµ amν + h.c.]. (1) n,m µ,ν
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Here, a†nµ (amν ) indicates the creation (annihilation) operator of µ(ν)-th atomic orbital at n(m)-th atom. Also, εnµ and tnµ,mν describe elements of on-site matrices of S and Mo atoms and elements of hopping matrices between first-nearest neighbor atoms, respectively. Using the Slater-Koster transformations, these elements of hopping matrices are derived from direction cosines and the related Slater-Koster tight-binding parameters [35]. The related parameters of pdσ and pdπ for the nearest-neighbors of Mo-S, ddσ, ddπ and ddδ for nearest-neighbor of Mo-Mo, and also ppπ and ppσ for nearest-neighbor of S-S are given in Table 1. The armchair and zigzag MoS2 nanoribbons can be specified by number of dimer lines Na and number of zigzag lines Nz across nanoribbon, respectively. An ideal MoS2 nanoribbon is constructed by repeating periodically unit-cell of nanoribbon only along one direction. Additionally, by taking account only interaction with first nearest-neighbor unitcells, the Hamiltonian in k space will be obtained as H(kx ) = Hl−1,l exp(−ikx Tx ) + Hl,l + Hl,l+1 exp(ikx Tx ).
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Here, Hl,l and Hl,l+1 describe the Hamiltonian of interactions between atoms of l-th unitcell and between atoms of l-th with atoms of l + 1-th unit-cell, respectively. Also, Tx indicates lattice constant of the nanoribbon. It should be noted that, the band structure is obtained from eigenvalue calculations of above matrix. Furthermore, Hl,l and Hl,l+1 matrices are 11N ×11N blocks, which N is Na (Nz ) in the armchair (zigzag) nanoribbon.
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The intrinsic Spin-orbit interaction
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In the presence of the ISOC, the Hamiltonian of the system can be given in the spin space by H = HSK ⊗ 12 + HISOC ,
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∆α Lα · Sα . h ¯
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HαISOC =
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where, as afore-mentioned in earlier subsection, HSK is the Hamiltonian system in the absence of the spin-orbit interaction. The second term represents the Hamiltonian of the intrinsic spin-orbit interaction, which for each site is given by
Here, ∆α is the ISOC constant, which depends on the type of specific atom (α = Mo and S). Also, L and S are orbital and spin angular momentum operators, respectively. Here, the spin-orbit interaction effect can be evaluated using the representation − − + ∆α L+ α Sα + Lα Sα α z z (5) HISOC = + Lα Sα h ¯ 2 where, +
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The orbital angular momentum operator L acts on |l, mi state as p L± |l, mi = h ¯ l(l + 1) − m(m ± 1)|l, m + 1i, Lz |l, mi = m¯h|l, mi. 4
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Here, l and m denote the quantum number of angular momentum and its z component, respectively. The chosen basis sets can be presented as 1 i |pz i = |1, 0i, |px i = − √ (|1, 1i − |1, −1i), |py i = √ (|1, 1i + |1, −1i), 2 2
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|d3z2 −r2 i = |2, 0i , |dx2 −y2 i = √12 (|2, 2i + |2, −2i), |dxy i = − √i2 (|2, 2i − |2, −2i), |dxz i = − √12 (|2, 1i − |2, −1i), |dyz i = √i2 (|2, 1i + |2, −1i).
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The ISOC term is taken into account quantum effects of interaction of spin with orbital motion of electron. The hopping matrices of the HISOC block (matrix elements of the spinorbit coupling Hamiltonian) between different atoms are assumed to be zero, because the spin-orbit interaction has its largest effect on carriers. So, the intra-atomic interaction is acted on all atoms which its value depends on the atom specie. The on-site matrices of the HISOC block act on the basis states |d3z2 −r2 i, |dx2 −y2 i, |dxy i, |dxz i and |dyz i √ √ 3 3 0 0 0 − i 0 0 0 0 0 2 2 0 1 1 0 0 −i 0 0 0 0 i 2 2 1 1 Mo Mo 0 0 −2i εISOC,↑↑ = 0 i 0 0 0 , εISOC,↑↓ = √0 2 , 0 0 0 0 −i − 12 21 i 0 0 √23 2 − 3 0 0 0 2i 0 i − 12 i − 12 0 0 2 (10) √ √ 3 3 0 0 0 i 0 0 0 0 0 2 2 0 1 1 0 0 i 0 0 i 0 0 − 2 2 0 1 1 Mo Mo 0 0 − i − εISOC,↓↓ = 0 −i 0 0 0 , εISOC,↓↑ = √ 2 2 . − 3 1 1 0 0 0 0 i i 0 0 2 2 2 2 √ i 3 − 1 1 0 0 0 −2 0 i −2i 2 0 0 2
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In a similar way, the resulting on-site matrices states |px i, |py i and |pz i as 0 − 21 i 0 0 S S 1 0 εISOC,↑↑ = 2 i 0 0 , εISOC,↑↓ = 0 0 0 − 12 1 0 i 0 0 2 S S 1 εISOC,↓↓ = − 2 i 0 0 , εISOC,↓↑ = 0 1 0 0 0 2
of the block HISOC act on the basis 1 0 2 0 − 12 i , 1 i 0 2 0 − 12 0 − 12 i . 1 i 0 2
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The Hamiltonian of the intrinsic spin-orbit interaction is given in the spin space by ↑↑ HISOC H↑↓ ISOC HISOC = . (12) ↓↓ H↓↑ ISOC HISOC So, the intra-atomic ISOC Hamiltonian becomes a 22N ×22N matrix, which N is Na (Nz ) in the armchair (zigzag) nanoribbon. 5
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2.3
The Rashba Spin-orbit interaction
Under applying an electric field, the Hamiltonian of the Rashba spin-orbit interaction in a continuum space is given by E ) ≡ tR z · (σ×k), (13) 2mc2 where, µB and σ are the Bohr magneton and spin Pauli matrices, respectively. In addition, the Rashba coefficient tR , indicates the strength of RSOC, which evidently is proportional to the gradient potential. Corresponding tight-binding Hamiltonian gets the form XX HRSOC = (itR c†n,µ [σ × dˆn,n+1 ]cn+1,ν ). (14) n
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Here, dˆn,n+1 describes the unit vector connecting of site n to site n + 1. The Hamiltonian of the Rashba spin-orbit interaction mixes different spins and gives rise to flip the spin of electron from up to down, which in the spin space takes the following form 0 H↑↓ RSOC HRSOC = . (15) H↓↑ 0 RSOC It should be added that, the inter-atomic RSOC Hamiltonian, 22N ×22N matrix, can be split into the hopping matrices acting on the basis states corresponding to first nearestneighbor atoms. The Rashba hopping integrals matrices are measured in the unit of corresponding Slater-Koster hopping matrices as (16)
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↓↑ t↑↓ nm,RSOC = βtnm , tnm,RSOC = γtnm ,
where β =p0.5itR cos(φ)[sin(θ)+i cos(θ)], and γ = 0.5itR cos(φ)[sin(θ)−i cos(θ)]. In which cos(φ) = 4/7 and also θ is the angle between in-plane nearest-neighbor atoms in the MoS2 monolayer [30].
Spin-polarized current
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Here, we investigate the electrical transport of the planar devices in the presence of Rashba spin-orbit interaction, with a combination of spatial symmetrical relations and time-reversal symmetry are considered. It was been explained in Ref. [36] and we discuss about these symmetrical relations at next section. However, the numerical studies can reveal strength of spin polarization of different systems under study. The device is assumed including central region in the presence of both types of spinorbit interactions and two semi-infinite leads in the absence of any type of spin-orbit coupling. The transmission of the device is computed within the NEGF method [37, 38]. The transmission probability of an incident electron through a device is obtained as a function of incident energy from the extended Green’s function of the central region, GC , as T (E) = Tr[ΓL G†C ΓR GC ].
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The Greens function of the central region coupled to the left and right electrodes is given by GC (E) = [(E + i)I − HC − ΣL (E) − ΣR (E)]−1 , 6
(18)
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where, is an arbitrarily small number added into the above equation to induce the convergence criterion and I is the identity matrix. Here, the broadening function ΓL(R) indicates the coupling between the device and the left (right) electrode. ΓL(R) = i[Σ†L(R) − ΣL(R) ].
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In above consideration, ΣL(R) is the self-energy of the left (right) electrode, and it is calculated from the surface Green’s function of the left (right) electrode and interaction between the central region and the left (right) electrode gL(R) via ΣL (E) = HLC † gL HLC , ΣR (E) = HCR gR H†CR .
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In order to calculate the surface retarded Green’s function of the semi-infinite left and right leads, we use the transfer matrix technique as [25, 39, 40]. −1
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(g0,0 )L = [(E + i)I − H0,0 − H−1,0 † Λ] ,
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(gM+1,M+1 )R = [(E + i)I − H0,0 − H−1,0 Λ† ] .
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Here, Λ and Λ are the transfer matrices, which can be efficiently calculated within an iterative scheme as defined by Refs. [25, 39, 40]. In Eq. (21), H0,0 represents the Hamiltonian of the surface unit-cell and H0,1 is its interaction with the adjacent supercells. With above consideration, we can calculate the TDOS of the system at the energy E as 1 TDOS(E) = − ImTr[GC (E)]. π
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Moreover, in the spin space, the spin-resolved transmission coefficient is given by 0
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σ σσ † σσ where Tσσ = Γσ0 Gσσ denotes transmission probability of electron with 11 Γ2 (G11 ) . T 0 spin σ through the central region when the electron with spin σ is injected into the region, in the presence of both spin-flip and spin-conserved processes. It can be rewritten in the spin space as
T↑↑ T↑↓ T↓↑ T↓↓
↑↑ ↑ ↑ −1 ΣR 0 H H↑↓ ΣL 0 − . (24) = (E + i)I − − H↓↑ H↓↓ 0 Σ↓R 0 Σ↓L
Computationally, the transmission T↑↑ and T↑↓ can be obtained, when one considers that, only spin-up electrons inject from the left electrode into the central region. Finally, the spin polarization along to z direction, (SP )z , is defined as (SP )z =
(T ↑↑ + T ↓↑ − T ↑↓ − T ↓↓ ) . (T ↑↑ + T ↓↑ + T ↑↓ + T ↓↓ )
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Calculated results and discussions
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The main purpose of the work is to investigate theoretically electronic properties of the AMOS2 and ZMOS2 nanoribbons in the absence and in the presence of the ISOC and also the RSOC effects. The whole of energy parameters has been scaled dimensionless. Also, c, e and h are set to 1. Besides, an in-plane interatomic spacing is taken 0.316 nm according to the experimental evidence [30]. Therefore, the lattice constant Tx of the AMoS2 and ZMOS2 nanoribbons are selected as 0.547 nm and 0.316 nm, respectively. The band structure of nanoribbons is calculated numerically using 50 k-points sampling along to Γ − X direction. Additionally, using Eqs. (18)-(22), the TDOS of the nanoribbons will be obtained in the presence of spin-orbit coupling, when all three regions (two electrodes and the central region) are affected due to them. Also, in all calculations, the length of central region is assumed to be one unit-cell. Since, the considered length for the central region is much smaller than 14-18 nm, which this length is the mean free path of MoS2 monolayer [41, 42]. So, the transport properties of the system are calculated in the ballistic regime in the absence of any scattering process. In addition, the arbitrarily small number, is taken 0.0002 eV. The ISOC constant for Mo (∆Mo ) and S (∆S ) atoms is taken 0.075 eV and 0.052 eV, respectively [31]. Let us start by investigating the influence of ISOC as well as electric field induced RSOC on the electronic properties of the MoS2 nanoribbons. The number of the dimer line of AMoS2 ribbon and the zigzag line of ZMoS2 ribbon is given by Na = 10 and Nz = 6, respectively as schematized in Figs. 1(a) and 2(a). These two nanoribbons possess nearly same width 1.420 nm and 1.457 nm, respectively. Figs. 1 and 2 show the band structure and TDOS of two types of MoS2 nanoribbons in the absence and also in the presence of two types of spin-orbit couplings. Generally, the figures clearly show that the electronic properties of MoS2 nanoribbons strongly depend on the type of nanoribbon with different edge terminations. Figs. 1(b) and 2(b) reveal that, both of the armchair and zigzag nanoribbons under study exhibit n−type semiconductor behavior, which possess the bandgap of about 1.56 eV and 1.04 eV, respectively. The spin-less electronic properties of the nanoribbons are discussed completely at greater length in Ref. [25]. Moreover, the dangling bonds of edge atoms induce some mid-gap states close to the conduction band in the AMoS2 ribbon, which these mid-gap states localize mainly on S atoms. While, two Mo edge atoms have dominant association to constitute top of the valence band in the AMoS2 nanoribbon (the figure is not shown). The origin of these can be related to the chemical nature of S and Mo atoms. Indeed, S atom is more electronegative with respect to Mo atom. So, this causes electrons to accumulate on S atoms at the edge which, acts as adatoms. Contrary to the armchair nanoribbon, the states due to only S (Mo) atoms at the edge constitute the bottom (top) of conduction (valence) band in the ZMoS2 nanoribbon, which this atomic contribution is unaffected due to the RSOC. In addition, Figs. 1(c) and 1(d) (2(c) and 2(d)) in comparison to Fig. 1(b) (2(b)) illustrate that, both of intrinsic and Rashba spin-orbit couplings lift band degeneracy and split bands of the AMoS2 (ZMoS2 ) nanoribbon into subbands. It is seen that the band spin splitting enhances when both of them are assumed. The reason of this splitting is explained as follows: existence of both time-reversal and spatial inversion symmetries in the system simultaneously gives rise to spin degeneracy of the band structure [11]. For example, an fcc lattice has an inversion symmetry. Therefore, the band degeneracy of this lattice (e.g., Au) is preserved in the presence of spin-orbit interaction. While, the spin 8
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degeneracy of bands is removed, when there is spin-dependent term in the Hamiltonian of a system without inversion symmetry. On the other hand, the time-reversal symmetry is preserved in the absence of an external magnetic field or magnetic impurities [36]. Also, the spin-orbit interactions lead to broadening of the density of states to some extent. Besides, Fig. 1(d) (2(d)) suggests that the RSOC with the Rashba coefficient tR = 0.1 does not cause the band gap to change noticeably in the AMoS2 (ZMoS2 ) ribbon. Moreover, the effects of Rashba coefficient intensity are studied on the electronic properties of MoS2 nanoribbons in Fig. 3. In this regard, the band structure and the TDOS of the AMoS2 and ZMoS2 nanoribbons are calculated for two different values of the Rashba coefficient, 0.1 and 0.5. The figure clearly demonstrates that the band spin splitting and also broadening of the TDOS enhance when the intensity value of Rashba coefficient increases. Evidently, the reason of this enhancement is related to the increase of spin-flip rate due to higher intensity value, which will be discussed more at next paragraph. Fig. 4 displays the spin-resolved transmission, spin polarization (SP )z and band structure of AMoS2 nanoribbon as a function of carrier energy in the presence of combination of the Rashba and also intrinsic spin-orbit couplings. Here, the numerical calculations are performed with tR = 0.5. By referring to the Fig. 4(a), we can see that the T ↑↓ and T ↓↑ appear noticeably in the presence of the RSOC. Indeed, the effect causes spin-flipping process to occur for the transmitted carriers through the channel. One should note that, T↑↓ and T↑↑ indicate spin-flipping and spin-conserving processes during carrier transport through Rashba device, respectively. According to Figs. 4(b) and 4(c), both of the incident electrons at the lowest subbands and the incident holes at the highest subbands around the Fermi level contribute to negative spin polarization (SP )z . So, by confining the electrons to occupy only these subbands, the spin-polarized current is generated and the system can be used as a spin filter device. However, the predicted spin-polarized current occurs due to the electric field induced RSOC in other quasi-one dimensional systems such as graphene nanoribbons and carbon nanotubes [36, 43]. Besides, the spin polarization of AMoS2 ribbon transmission close to the Fermi level is originated mostly from the edge and mid-gap states. Moreover, as shown in the panels (c)-(e) of Fig. 4, the spin splitting of the bands increases significantly, when the RSOC is combined to the inherent coupling. In order to gain a better understanding of the effect of the RSOC on the transport properties of zigzag ribbons, we have calculated the spin-dependent transmission, spin polarization (SP )z and band structure of ZMoS2 ribbon as a function of the incident electron energy in Fig. 5. From comparison Figs. 4a to 5a, we can see that the spinflipping process for the ZMoS2 ribbon occurs less than the AMoS2 ribbon particularly close to the Fermi level. Intriguingly, the RSOC does not introduce a noticeable spin polarization in the ZMoS2 ribbon even close to the Fermi level. While, the lowest subband of the incident electrons in the ZMoS2 ribbon splits significantly, when the RSOC with tR = 0.5 is combined to intrinsic one (see Fig. 5(c)). The reason may be related to construction of the lowest and highest subbands, which originates from only Mo and S edge atoms. Therefore, one can conclude that the S and Mo edge atoms do not have any effect to generate a spin-polarized system based on the zigzag nanoribbon under study in the exposure to the electric-field induced RSOC. In order to explore the influences of edge termination of the Rashba devices based on two types of nanoribbons, we investigate the spin polarization of several different widths and edge terminations: Na = 8, Na = 10 and Na = 12 for AMoS2 , and Nz = 5, Nz = 6 9
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and Nz = 7 for ZMoS2 ribbons. Fig. 6 clearly shows that the induced spin polarization of ZMoS2 nanoribbons is much smaller than that of the considered armchair ones. Besides, the position of energy peaks depends to large extent on the type of ribbon. Also, the most important difference between spin polarization of different armchair nanoribbons close to the Fermi level is that the contribution of incident electrons at the lowest conduction subbands changes from negative spin polarization to positive spin polarization, when the dimmer lines varies from Na =10 to Na =12. Fig. 7 shows the band gap evolution of two types of MoS2 ribbons due to the RSOC with different Rashba coefficient. The RSOC induces gradual decrease in the band gap of the AMoS2 and ZMoS2 ribbons when the Rashba coefficient increases. Whereas, the slop of variation is nearly same for both nanoribbons under study. Finally, we examine the effect of a uniform exchange field on the spin polarization of MoS2 ribbons in the presence of the RSOC. The exchange field may be originated from the proximity with a ferromagnetic insulator substrate or from depositing ferromagnetic atoms at the edge of ribbon. Hence, this breaks the time-reversal symmetry. The Hamiltonian of the Rashba device [28] in the central region becomes X a†nµ σz anµ , (26) H0 = H − M n,µ
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when the exchange field is taken into account. In which, M indicates the strength of the exchange field. In Fig. 8, we depicted the spin polarization of AMoS2 and ZMoS2 nanoribbons in exposure to combination effects of the intrinsic and the RSOC and also a uniform exchange field. The numerical calculations are carried out for tR = 0.5 and M = 2.4∆α . Generally, the exchange filed causes the spin polarization to increase in the studied nanoribbons to some extent, which this increase is more pronounced close to the Fermi level. Despite to the RSOC, which it can not introduce the noticeable spin polarization in the ZMoS2 , combination of the exchange field and the RSOC induce spinselective current especially close to the Fermi level. Such that, the spin polarization of ZMoS2 reaches 100% via considering the exchange field close to the Fermi level.
Our main focus in this work was the role of the electric field induced Rashba spin-orbit interaction in introducing spin-resolved transport in the planar devices based on MoS2 nanoribbons. The results show that, regardless of significant spin splitting in the lowest conduction bands of the zigzag MoS2 nanoribbons, the Rashba-type spin-orbit interaction cannot induce significant spin polarization in the output electrode of a two-terminal system based on the planar zigzag MoS2 ribbons with about width of 1.457 nm and different edge terminations. While a net spin polarization in the armchair MoS2 ribbons with nearly same width is provided mostly close to the Fermi level which, the predicted net spin polarization is originated mostly from edge states. Whereas, the induced spin polarization enhances, when applying a vertical electric field is combined to proximity of a ferromagnetic insulator substrate. Such that, the spin polarization of ZMoS2 (6) reaches 100% close to the Fermi level via considering the exchange field. Therefore, the tunable spin-dependent transport can be provided in the MoS2 nanoribbons.
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Acknowledgment
This work has been supported by Yasooj Branch, Islamic Azad University through a research fund.
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Figure 1: (a) Schematic representation of AMoS2 nanoribbon with width of 1.420 nm, the band structure and the TDOS in (b) the absence and in the presence of (c) the ISOC and also (d) the RSOC with the tR = 0.1. The unit-cell of the nanoribbon is composed of 30 atoms and is identified by the black line. The red (gray) circle denotes the S (Mo) atom.
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Figure 2: (a) Schematic representation of ZMoS2 nanoribbon with width of 1.457 nm, the band structure and the TDOS in (b) the absence and in the presence of (c) the ISOC and also (d) the RSOC with the tR = 0.1. The unit-cell of the nanoribbon is composed of 18 atoms and is identified by the black line. The red (gray) circle denotes the S (Mo) atom.
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Figure 3: The dependence of the band structure, TDOS as a function of the Rashba coefficient for (a) AMoS2 nanoribbon with Na = 10 and tR =0.1 and (b) tR =0.5 and also for (c) ZMoS2 nanoribbon with Nz = 6 and tR =0.1 and (d) tR =0.5. Figure 4: (a) Spin-resolved transmissions, (b) spin polarization with respect to the z direction, (SP )z and (c) band structure of AMoS2 nanoribbon with Na = 10 under a combined effect of the RSOC and the ISOC with tR = 0.5. Also band structure of AMoS2 nanoribbon (d) in the presence and (e) in the absence of the ISOC is presented.
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Figure 5: (a) Spin-resolved transmissions, (b) spin polarization with respect to the z direction, (SP )z and (c) band structure of ZMoS2 nanoribbon with Nz = 6 under a combined effect of the RSOC and the ISOC with tR = 0.5. Also band structure of ZMoS2 nanoribbon (d) in the presence and (e) in the absence of the ISOC is presented.
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Figure 6: Spin polarization, (SP )z , as a function of energy for AMoS2 nanoribbon with (a) Na = 8, (b) Na = 10, (c) Na = 12 and for ZMoS2 nanoribbon with (d) Nz = 5, (e) Nz = 6 and (f) Nz = 7 in the presence of the RSOC and the ISOC. In all cases, the Rashba coefficient is tR = 0.5. Figure 7: The dependence of the band gap of AMoS2 (Na = 10) and ZMoS2 (Nz = 6) on the Rashba coefficient. Figure 8: Spin polarization of (a) AMoS2 (Na = 10), (b) ZMoS2 (Nz = 6) nanoribbons in the presence of the RSOC and also the exchange field for M = 2.4∆α . The Rashba coefficient of tR = 0.5 is taken. Table 1: εMo 1 -1.512 ppσ 0.696
The Slater-Koster parameters for MoS2 monolayer are given in unit of eV [31]. εMo εMo εMo εMo εS1 εS2 εS3 2 3 4 5 -3.025 -3.025 0.419 0.419 -1.276 -1.276 -8.236 ppπ ddσ ddπ ddδ pdσ pdπ 0.278 -0.933 -0.478 -0.442 -2.619 -1.396 13
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• Two types of defect-free MoS2 nanoribbons have been investigated. • The vertical electric field and edge states in the system induce spin-resolved currents. • The creation and manipulation of the spin current by electrical means is a route to design the spintronics devices for low-consumption and high-speed operation. • Our calculations are based on NEGF within multi-band tight-binding method.