Effects of line defects on spin-dependent electronic transport of zigzag MoS2 nanoribbons

The nonlinear spin-dependent transport properties in zigzag molybdenum-disulfide nanoribbons (ZMNRs) with line defects are investigated systematically using nonequilibrium Green’s function method combined with density functional theory. The results show that the line defects can enhance the electronic transfer ability of ZMNRs. The types and locations of the line defects are found critical in determining the spin polarization and the current-voltage (I-V) characteristics of the line defected ZMNRs. For the same defect type, the total currents of the ribbons with the line defects in the centers are lager than those on the edges. And for the same location, the total currents of the systems with the sulfur (S) line defect are larger than the according systems with the molybdenum (Mo) line defect. All the considered systems present magnetism properties. And in the S line defected systems, the spin reversal behaviors can be observed. In both the spin-up and spin-down states of the Mo line defected systems, there are obvious negative differential resistance behaviors. The mechanisms are proposed for these phenomena.


I. INTRODUCTION
As one of the typical two-dimensional (2D) nanomaterials, graphene is a promising material for next-generation flexible electronic devices, [1][2][3] but the lack of a bandgap hampers its application in semiconducting devices.Recently, experiment and theory researches have found that the graphenelike single-layer transition metal dichalcogenide molybdenum disulphide (MoS 2 ) has a large intrinsic direct bandgap about 1.9 eV, [4][5][6] which makes it a very interesting substitute/complement to graphene in semiconducting and optical applications.For example, the transistor fabricated with MoS 2 atomic thin layers exhibits high channel mobility (µ > 200 cm 2 V −1 s −1 ) and excellent on / off current ratio (1 × 10 8 ). 7,8And in the single-layer based phototransistor, the photoresponsivity can reach as high as 7.5 mA/W, which is better than that obtained from the single-layer graphene based field-effect transistor (1 mA/W). 9t has been shown that the electronic properties of single-layer MoS 2 can be tuned by doping, 10 bending, 11 strain, 12 or defects. 13In practice, like any other 2D materials, the defects are inevitable in MoS 2 . 13,146][17] There are various types of structural defects in MoS 2 , such as point defects, 13 dislocations, 18 grain boundary, 19 or topological. 20The point defect is one of the native defects which have been studied both in experiment 13 and theory. 21,22 et al. used five different types of vacancy defects (namely, single Mo or S vacancy, S-S or Mo-S vacancy, and S-Mo-S vacancy) to modify the electronic and magnetic states of the single-layer MoS 2 honeycomb structures, and found that the system with S-Mo-S vacancy defects will gain magnetic properties. 23Zheng et al. predicted by first-principles calculations that macroscopic ferromagnetism of MoS 2 nanosheets could be introduced by applying doping vacancy defects. 24Cai et al. found that the transformation of the surrounding 2H-MoS 2 local lattice into a trigonal MoS 2 could be prompted if introducing S vacancy in a 2H-MoS 2 ultrathin nanosheet host.Then, a robust intrinsic ferromagnetic response may be created. 20ine defect is also another typical structural defect in MoS 2 .Yong at al. reported that a finite S atomic line vacancy on MoS 2 surface could behave as a pseudoballistic wire for electron transport. 25ifferent growth conditions may format different defects in the MoS 2 .Enyashin et al. observed the line defects in MoS 2 layers grown by exfoliation experimentally by atomic resolution transmission electron microscope (TEM).And using molecular dynamics, they identified two antiparallel MoS 2 parts by either Mo-Mo bonds or by -S-bridges. 26The same kinds of line defects have also been studied for MoSe 2 and WS 2 . 27,28From the samples grown by the chemical vapor deposition method, Zhou et al. found some other grain boundaries (GBs), such as 4|4P 60 0 GB, 4|4E 60 0 GB and 5|7 GB.And they could affect the properties of MoS 2 . 13,294][35] To the perfect ZMNRs, the atoms on the two edges are different, that is the Mo atoms on one edge, and the S atoms on the other.And in this case, the odd number (namely, asymmetry system) or even number (namely, symmetry system) of the width of the ZMNRs has very little effect on their electronic structures and transport properties. 36,37It is different from the zigzag graphene nanoribbons (ZGNRs), in which the electronic transport properties depend on the symmetry of the edges strongly.And only in the asymmetry ZGNRs, the obvious spin-dependent electronic transport properties occur. 38,39n order to study the effects of the edge atoms on the spin-dependent electronic transport properties for ZMNRs, in this paper, according to the experimental results of Enyashin et al., 26 we introduce different line defects in the nanoribbons, including Mo line and S line defects.Then, the edge atoms of ZMNRs are the same for each system.And based on the first-principles calculations, we present a unique and intrinsic electronic transport property for these line defected ZMNRs under finite bias voltage.

II. COMPUTATIONAL DETAILS
In the calculations, the considered device systems as illustrated in Fig. 1: M0 (the perfect ZMNR), M1 (ZMNR with Mo line defect in the center), M2 (ZMNR with S line defect in the center).And, in order to predict the effects on the electronic properties of the ZMNRs by the location of the line defects, we also proposal two other defected systems: M3 (ZMNR with Mo line defect on the edge) and M4 (ZMNR with S line defect on the edge).
The geometrical optimizations and charge density calculations of the ZMNRs unit cells are based on the Vienna ab inition Simulation Package (VASP) with a projector augmented wave (PAW) scheme. 40,41The vacuum layers are 15 Å to avoid the artificial Coulomb interactions between the contents in two neighboring cells.In the calculations, a plane wave cutoff energy of 400 eV and a Monkhorst-Pack k-point sampling of 1 × 1 × 21 with a convergence criterion of 10 −5 eV, and the maximum Hellmanm-Feynman forces acting on each atom is less than 0.05 eV/Å upon ionic relaxation, are chosen to achieve the balance between calculation efficiency and accuracy.The spin-dependent electronic transport properties are calculated based on ATK (Atomistix Toolkit) package, 42,43 in which the real-space, nonequilibrium Green's function (NEGF) formalism and density functional theory (DFT) are implemented.Although according to Zhang et al., 44,45 the method based on steady-state DFT would be more accurate for the current calculation than the method based on DFT+NEGF.But they yield very similar I-V (current-voltage) curves.And the difference between the both methods is very small when the bias is larger than 0.2 V.So the method based on ground state DFT is used in our calculations, in which the bias is between 0 V and 1 V.
The two-probe systems, shown in Figs.1(a)-1(e), include three regions: left electrode, right electrode and the central region, in which the electrodes are two semi-infinite ZMNRs.In the calculations, the local spin density approximation (LSDA) is adopted as the exchange-correlation function and the single-zeta plus polarized (SZP) basis set is used.The parameters set as follows: cutoff energy of all atoms is 150 Ry, k-point grid is 1 × 1 × 500, and the electronic temperature is 300 K.And the NEGF-DFT self-consistence is controlled by a numerical tolerance of 4 × 10 −5 eV.To obtain accurate results, all the considered two-probe systems have performed geometry optimization until the force on each atom is less than 0.05 eV/Å.The corresponding spin-dependent current I σ (V b ) through the central region under external bias voltage (V b ) is calculated using the Landauer-Büttiker formula 46 where e is the electron charge, h is the Planck's constant, σ represents spin-up (↑) or spin-down (↓), where  R l /r is the retarded self-energy matrix which takes into account the left/right electrode, G R/ A is the retarded/advanced Green's function of the central region.

III. RESULTS AND DISCUSSIONS
Firstly, we plot the calculated spin-dependent transmission spectra T(E,V b ) at zero bias voltage for the four line defected systems M1-M4, respectively, as shown in Figs.2(b)-2(e).And, as a comparison, the T(E,V b ) of perfect system M0 has also been shown in Fig. 2(a).It can be seen clearly that all the spin-up and spin-down states are non-degenerate in the five systems, illustrating the magnetism characteristics of them.As we know, the perfect ZMNRs have magnetism characteristics. 14,36hen, the line defects in the ZMNRs don't remove the magnetism characteristics of them.And by calculations, we obtain the magnetic moments of the unit cell (shown in Fig. 1) for M0-M4 are 1.2275µ B , 3.1598µ B , 1.2562µ B , 2.1255µ B and 1.0121µ B , respectively.From the calculated spatial spin distribution (up-down), as shown in Figs.3(a)-3(e), we can see that all the five systems depict ferromagnetism properties.In the both edges of each system, the magnetisms of spin-up states are larger than that of the spin-down states.And for M0, M3 and M4, the spin distribution is almost on the edge Mo or S atoms, minor on the next edge atoms, and very little on the central atoms.While for M1 and M2, when the line defect is located in the center, though the magnetism mainly concentrates on the edge Mo or S atoms, we can clearly see that there a part of spin density distributes on the atoms in or near the center.As a result, the line defects in the center will strengthen the magnetism of the ZMNRs.That's why the magnetic moment of M1 is larger than M3, and M2 is larger than M4.
To explore the different effects of the line defects on the transport properties of ZMNRs, in Figs.4(a under zero bias for the spin-up and spin-down states of M0-M4.The transmission pathway is an analysis option which splits the transmission coefficient into local bond contributions, T ij .For example, if a system is divided into 2 parts A and B , the pathways across the boundary between A and B sum up to the total transmission coefficient 47 And in the pictures, the arrow indicates the direction of the electron flow, the volume and color of each arrow designate the magnitude of the local transmission.We can find that two different kinds of transmission channels can be seen in the systems: electron transmission via a chemical bond (such as Mo-S bond) and through hopping between atoms (such as Mo and Mo atoms, S and S atoms).phenomenon has also seen in GNR-based and H terminated MNR-based devices. 48,37From Fig. 4(a), we can see that the electrons of the pristine ZMNR (M0) flow mainly through the edges of ribbons and reach the other electrode.And for the defected systems M1-M4 (shown in Figs.4(b)-4(e)), the electrons flow not only through the edges of ribbons, but also through the atoms around the line defects.Especially for M1 and M2, in which the line defects in the center of the ribbons, the electron transmission pathways are mainly in the defected atoms.So, the line defects can enhance the electronic transfer ability of ZMNRs.Moreover, if the line defects are in the center of the ribbons, the electronic transfer ability will enhanced greatly.We consider that in the positions where the line defects are, the Mo or S atoms are unsaturated, and then there are new edge states introduced into the sides of the line defects for the ribbons.Thus, in the line defected systems, there are more transport channels for the electrons than that in the perfect ZMNRs.Meanwhile, we have found that the bond lengths of Mo-S, Mo-Mo and S-S near the line defects are shorter more or less than those in the perfect system.Then the shorter bond lengths may make the electron transport more fluently.Thus, the electron transport ability is enhanced.
In the following, we simulate the spin-dependent transport properties of the line defected systems M1-M4 and the pristine system M0.Their spin-dependent I-V curves under the bias range of [0 V, 1.0 V] are presented in Fig. 5(a).It is shown that both the currents of spin-up (I ↑) and spin-down (I ↓) states are split in all the systems.So, all the five systems have magnetism characteristics.We also find that there is no threshold voltage both for the spin-up and spin-down states, they all display metallic characteristics.And all these currents are in good accordance with the transmission spectra shown in Fig. 2. The I-V curves are evidently dependent on the types and locations of the line defects.From the total currents of the five systems (shown in Fig. 5(a')), we can clearly see that under the same voltage, the currents of all defected systems are larger than that of the perfect one, so the electronic transfer ability of the devices has been strengthen when the line defects are introduced in the ZMNRs.Comparing M1 with M3, M2 with M4, we can find that I M1 > I M3 , and I M2 > I M4 under the same bias voltage.Thus, for the same defect type, the more the line defects close to the center of the ribbons, the better for the currents.And for the same location, the current of the system with the S line defect (M2 or M4) is larger than the according system with the Mo line defect (M1 or M3).
Interestingly, for M1 and M3, the current of spin-up state is almost larger than that of the spindown state under the whole considered bias region, but for M0, M2 and M4, the value of I ↑ −I ↓ includes both positive and negative values with the increasing of the bias.Thus, the current polarization of the defected systems can be controlled by the bias voltage and the sign-switching behavior is shown in the three systems.And this reversal of spin-polarization caused by bias voltage maybe has potential application in logic spintronics devices.To quantify the spin-polarization behavior, we define the spin-polarized effect (SPE) η by the following formula, η = I ↑−I ↓ I ↑+I ↓ × 100%.The calculated SPEs of the five systems are shown in Fig. 5(b).We can see that the absolute maximum SPEs are 24% at 0.3V, 33% at 0.8 V, 12% at 0.1 V, 37% at 0.3 V, and 22% at 0.8 V for M0, M1, M2, M3 and M4, respectively.
To understand the origin of the reversal of SPEs, take M4 as an example, we study the transmission spectra, local density of states (LDOSs) at Fermi level of M4 at 0.4 V and 0.8 V in Figs.6(a) and 6(b), respectively.From Eq.(1), we know that [µ l (V b ), µ r (V b )] refers to the energy region which contributes to the current integration, and it is called bias voltage window or integration window.If the Fermi level is set to zero and bias voltage window is [−V b /2,V b /2], the current is determined by T(E,V b ) in the bias voltage window.We can see from Fig. 6(a) that in the bias window, the integration area of the spin-up is larger than that of the spin-down state.And the corresponding LDOS of spin-up electron is more delocalized and distributes over the edges of the ribbon.Both the contrasts mentioned above indicate that the spin-up electron dominates over the spin-down electron for M4 at V b = 0.4 V, in according with the positive SPE about 19% (shown in Fig. 5(b)).However, the dominance between the spin-up and spin-down electrons is overturned at V = 0.8 V (Fig. 6(b)), where the comparison of transmission spectra, as well as the LDOSs, indicates a dominance of the spin-down electron over the spin-up electron, in agreement with a negative SPE of 22%.Thus the changing dominance of spin-up and spin-down from V b = 0.4 V to 0.8 V can be attributed to the sign-changing of the SPEs in M4.
Furthermore, as shown in Fig. 5(a'), the currents increase with the bias both in M2 and M4.Nevertheless, the currents decrease with the bias in some bias regions for M0, M1 and M3, thus the obvious negative differential resistance (NDR) behavior can be observed in the perfect ZMNRs and the systems with Mo line defects.The NDR property is very important in the field of electronic technology.It has a wide range of potential applications in digital application, amplification, and oscillators.However, the origin of the mechanism for NDR is still under debate. 49,50To understand the underlying mechanism of the NDR behavior, take M1 as an example, we calculate the transmission spectra as a function of electron energy and the projected density of states (PDOS) of central scattering region at 0.3 V and 0.7 V, as shown in Figs.7(a)-7(d).From Figs. 7(a) and 7(b), we can see that there is a strong transmission peak appears around Fermi level at 0.3 V.However, it becomes lower when the bias voltage is 0.7 V.And it can be seen that the integration area under 0.3 V is much larger than that under 0.7 V.The PDOS curves shown in Figs. this area.In order to give more intuitive electron transmission picture, the transmission eigenstates at the transmission peak for M1 under these two biases are plotted separately over Figs.7(a) and 7(b).It clearly shows that the wave function distribution of transmission eigenstates under 0.3 V is delocalized over the both edges of the ribbon.However, the wave function distribution is rare and localized under 0.7 V. Especially on the bottom edge of the ribbons, there is almost no wave function distribution of transmission eigenstates.So the current through M1 will decrease when the bias goes from 0.3 V to 0.7 V and the NDR behavior appears.

IV. CONCLUSION
We have presented the first-principles theoretical investigations on the spin-dependent electronic transport properties of ZMNRs with line defects in the central or on the edge of the nanoribbons.Compared with the perfect ZMNR, line defects will strengthen the electronic transport ability of the systems.For the same defect type, the more the line defects close to the center of the ribbons, the better for electron transport ability.And for the same location, the electron transport ability of the system with the S line defect is larger than the according system with the Mo line defect.And in the S line defected systems, there are spin reversal behaviors can be observed.Moreover, the NDR behaviors are predicted in the perfect ZMNRs and the ZMNRs with Mo line defects, which originate from the changes of PDOS peaks and wave function distribution of transmission eigenstates at different bias.All these characters would be potentially useful in the fabrication of spin electronic devices based on ZMNRs.
are the chemical potential of the left and right electrodes, f (E − µ l /r ) is the Fermi-Dirac distribution function of the left(l)/right(r) electrode, and T σ (E,V b ) is the quantum mechanical transmission probability of electrons, defined as

015015- 6 Li
FIG. 5. (a)The spin-dependent current, and (b) the SPEs as function of the applied bias voltage for M0-M4.The ↑ and ↓ refer to spin-up and spin-down states, respectively.The insert panel (a') represents the total current of M0-M4.

FIG. 6 .
FIG. 6.The transmission spectra and LDOSs for M4 at (a) V b = 0.4 V, (b) V b = 0.8 V.The regions between the blue dash lines are the bias windows.The ↑ and ↓ refer to spin-up and spin-down states, respectively.The zero of energy is set to Fermi level.The isovalue is 0.5 Å −3 .

8 Li
FIG. 7. The transmission spectra and PDOSs of M1, (a) and (c) for 0.3 V, (b) and (d) for 0.7 V.The pictures up (a) and (b) present the transmission eigenstates at the transmission peaks marked by the arrows.The red lines refer to the bias window, and the zero of energy is set to Fermi level. Ataca