Magnetic Behavior in TiS3 Nanoribbon

The electronic structure, magnetic properties and strain response of N-a-TiS3 nanoribbons are investigated by first-principles calculations. We find that the magnetic ground state is strongly dependent on width of a-TiS3. When N equals an odd number the ground state is a ferromagnetic (FM) metal, meanwhile, when N equals an even number the ground state is an anti-ferromagnetic (AFM) metal. More interestingly, a tensile strain as large as 6% can tune the 9-a-TiS3 nanoribbon from a FM metal to a half metal. A 4% tensile strain also causes a phase transition from AFM to FM ground state for 10-a-TiS3 nanoribbon. Our findings show that N-a-TiS3 is a promising candidate for spintronic and electronic applications.

Very recently, a new two-dimensional (2D) material, namely TiS 3 , has been successfully synthesized [25][26][27][28]. It is reported that monolayer TiS 3 exhibits a direct band gap of 1.1-1.2 eV [29], making the bandgap of TiS 3 comparable to that of silicon (1.1 eV). Also, numerical results show that the bandgap of TiS 3 is quite robust, almost independent of layer thickness, vertical strain and stacking order [30]. Moreover, a recent theoretical study proposed that TiS 3 is expected to have a higher electron mobility of 10,000 cm 2 V −1 s −1 [29]. Such a robust bandgap and ultrahigh electron mobility make TiS 3 a good candidate for nanoelectronics and optoelectronics application. The elastic modulus, carrier mobility and band structure of TiS 3 are strongly anisotropic due to its highly anisotropic crystal. Experimentally, nanoribbons of TiS 3 were successfully prepared [25][26][27][28]. Although, there are several studies on nanoribbons of TiS 3 [25][26][27][28][31][32][33], the magnetism of TiS 3 nanoribbon still lacks systematic study.
In this article, using first-principles calculation, we systematically study the magnetic properties of TiS 3 nanoribbon. Our results show that a-TiS 3 NR is a spin-polarized metal with local magnetic moments at the edge, while b-TiS 3 NR is nonmagnetic semiconductor. More interestingly, the magnetic ground state of N-a-TiS 3 NRs (N = 7, 8,9,10,11) are width dependent. When N is odd, the ground state is ferromagnetism (FM), meanwhile the ground state is anti-ferromagnetism (AFM)when N is even. This magnetic behavior is very different from other magnetic nanoribbons, such as zigzag black phosphorene nanoribbons [22] and zigzag graphene nanoribbons [12,13], whose magnetic properties are almost independent on the width of nanoribbon. We also find that compressive strain almost does not change the magnetic and electric properties of a-TiS 3 NRs. However, according to our calculations in 9-a-TiS 3 NR and 10-a-TiS 3 , tensile strain can change the magnetic and electric properties dramatically. 9-a-TiS 3 NR becomes a half metal under a tensile as large as 6%, and the AFM ground state of 10-a-TiS 3 is tuned into a FM ground state at about 4% tensile strain. Our findings show that the a-TiS 3 NRs are promising candidates for spintronics and electronics application.

Methods
All the calculations were carried with Vienna ab initio Simulation Package (VASP, version 5.3.2, University of Vienna, Vienna, Austria) [34,35]. The frozen-core projector augmented wave (PAW) method and the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) [36] were adopted. Cutoff energy was set as 500 eV for plane-wave expansion of the electronic wave function, and appropriate k-point meshes (9 × 1 × 1) were used for geometric optimization and self-consistent calculation. All structures were relaxed until the force on each atom was less than 0.01 eV/ Å. The energy convergence criteria were set to 1.0 × 10 −6 eV. A vacuum spacing of 15 Å was used so that the interaction in the non-periodic directions can be neglected. Since both Ti and S are not heavy elements, spin-orbit coupling (SOC) is expected to be small, thus the spin-orbit interaction is not included in all calculations.

Results and Discussion
The unit cell of monolayer TiS 3 is a rectangle. Our optimized lattice constants are a = 5.02 Å, b = 3.40 Å which are in good agreement with previous experimental result (4.96 and 3.40 Å) [37], and theoretical result (5.02 and 3.41 Å) [31]. There are two particular ways to cut monolayer TiS 3 into nanoribbons (cutting along the a axis or b axis). Then, two types of nanoribbons are indicated as N-a-TiS 3 NR and N-b-TiS 3 NR, where N indicates the number of Ti atoms in the unit cell of ribbon and a-TiS 3 NR and b-TiS 3 NR are along the a and b lattice vectors, respectively. Our calculations show that N-b-TiS 3 NRs are semiconductors without any local magnetic moment, which is consistent with previous theoretical results [31]. Thus, in the following discussion we will eliminate the N-b-TiS 3 NRs and mainly consider the N-a-TiS 3 NRs.
Here, we take N = 7-11 for a-TiS 3 NRs. Figure 1 shows the crystal structure of 10-a-TiS 3 NR and 9-a-TiS 3 NR. In particular, edge atoms may influence the physical properties. For a-TiS 3 NR, the edge atoms are S and Ti with dangling bonds which are local states around Fermi level as shown later. However, there are some differences in crystal structure between 9-a-TiS 3 NR and 10-a-TiS 3 NR. Dashed lines in Figure 1a,b also indicate the TiS 3 chain in monolayer TiS 3 . In 9-a-TiS 3 NR, the terminal atoms for every TiS 3 chain are the same, namely both of two edge atoms are Ti or S atoms for the same chain as shown in Figure 1b. However, in 10-a-TiS 3 NR, the edge atoms for every TiS 3 chain are different, i.e., one terminal atom is Ti and the other is an S atom shown in Figure 1a. Moreover, for N equal to all other odd numbers (or even numbers) the edge atom arrangement is the same as the N = 9 (or 10). This different edge atom arrangement will cause a different magnetic ground state of two NRs as discussed below. To clarify the effect of edge atoms, we plot the charge density isosurfaces of these three bands which cross the Fermi energy and the results are shown in Figure 2i-l. These pictures clearly illustrate the charge density is mainly localized at edge S and Ti atoms which again proves that three bands denoted by red lines are indeed the edge states. As we discussed before, the arrangement of edge atoms is different between 9-a-TiS3 NR and 10-a-TiS3 NR. Thus, charge distribution is different for these two a-TiS3 NRs. In 9-a-TiS3 NR, the charges are mainly localized at edge S atoms which are in the TiS3 chain, whose the end atoms are both sulfur atoms, shown in Figure 2i,j. The p orbital shape of charge density on edge S atoms are the hybridization of y p and z p unbonding state. However, the contribution of Ti-d state in TiS3 chain with two end Ti atoms is more complicated. The charge distribution shows that all Ti atoms in this chain have almost equal contributions to the three red bands. This means that the hybridization between Ti atoms is very strong. However, there is a very different charge distribution in 10-a-TiS3 NR. As we discussed previously, the terminals of TiS3 chain were one S atom and one Ti atom, rather than the two same atoms. Thus, the charge distribution of 10-a-TiS3 NR is very different from that of 9-a-TiS3 NR. In Figure 2k,l, we can see that there are p orbital shapes of charge density around edge S atoms, as in 9-a-TiS3 NR. More interestingly, the charge distribution on Ti atoms in the same chain are not equal at all. The biggest contribution comes from the edge Ti atom, and then the charge density decreases from the edge Ti atom to other side of NR which is different from the situation in 9-a-TiS3 NR. As we discuss later, this difference would cause the different magnetic ground states. Firstly, we calculate the electronic structure of 9-a-TiS 3 NR and 10-a-TiS 3 NR under GGA approximation without any spin polarization. The electronic structures of two nanoribbons are shown Figure 2. Three bands, denoted by red color, cross the Fermi energy indicating the metallic behavior of a-TiS 3 NR (shown in Figure 2a,e). Further analysis shows that these three bands are the edge states. Density of states (DOS), shown in Figure 2, indicate that three bands crossing the Fermi level are mainly contributed by the unbonding p y and p z states of edge S atoms and d z 2 and d x 2 −y 2 of edge Ti atoms. A peak of DOS near the Fermi energy, as shown in total DOS, indicates the possibility of spin polarization driven by Stoner instability [38].
To clarify the effect of edge atoms, we plot the charge density isosurfaces of these three bands which cross the Fermi energy and the results are shown in Figure 2i-l. These pictures clearly illustrate the charge density is mainly localized at edge S and Ti atoms which again proves that three bands denoted by red lines are indeed the edge states. As we discussed before, the arrangement of edge atoms is different between 9-a-TiS 3 NR and 10-a-TiS 3 NR. Thus, charge distribution is different for these two a-TiS 3 NRs. In 9-a-TiS 3 NR, the charges are mainly localized at edge S atoms which are in the TiS 3 chain, whose the end atoms are both sulfur atoms, shown in Figure 2i,j. The p orbital shape of charge density on edge S atoms are the hybridization of p y and p z unbonding state. However, the contribution of Ti-d state in TiS 3 chain with two end Ti atoms is more complicated. The charge distribution shows that all Ti atoms in this chain have almost equal contributions to the three red bands. This means that the hybridization between Ti atoms is very strong. However, there is a very different charge distribution in 10-a-TiS 3 NR. As we discussed previously, the terminals of TiS 3 chain were one S atom and one Ti atom, rather than the two same atoms. Thus, the charge distribution of 10-a-TiS 3 NR is very different from that of 9-a-TiS 3 NR. In Figure 2k,l, we can see that there are p orbital shapes of charge density around edge S atoms, as in 9-a-TiS 3 NR. More interestingly, the charge distribution on Ti atoms in the same chain are not equal at all. The biggest contribution comes from the edge Ti atom, and then the charge density decreases from the edge Ti atom to other side of NR which is different from the situation in 9-a-TiS 3 NR. As we discuss later, this difference would cause the different magnetic ground states. The previous theoretical study has proposed that the a-TiS3 NRs have local magnetic moment [31]. But detailed studies of magnetic properties are still lacking. Thus, we perform the spin-polarized calculation to find the magnetic properties of N-a-TiS3 NRs.
To explore the magnetic ground state of N-a-TiS3 NRs, we set two magnetic configurations, one is a FM arrangement with all magnetic moments in the same direction and the other is an AFM configuration with the magnetic moments of the two edges anti-parallel. From the results of the total energy calculation, we can easily find that the ground state of N-a-TiS3 NRs are not all FM which is different from the previous theoretical results [31]. The magnetic ground state varies as the width changes. Figure 3a shows that the energy difference between the magnetic state and nonmagnetic state, one can easily see that when N is odd, the ground state is FM, meanwhile N is even, the ground state is AFM. This width-dependent magnetic ground state is different from other 2D material NRs like black phosphorus [23].
Our calculations also reveal that all N-a-TiS3 NRs have metallic behavior with bands crossing the Fermi level. The electronic band dispersion of ground magnetic states of 9-a-TiS3 NR and 10-a-TiS3NR are shown in Figure 3b,c, respectively. For FM ground state of 9-a-TiS3, there is a spin The previous theoretical study has proposed that the a-TiS 3 NRs have local magnetic moment [31]. But detailed studies of magnetic properties are still lacking. Thus, we perform the spin-polarized calculation to find the magnetic properties of N-a-TiS 3 NRs.
To explore the magnetic ground state of N-a-TiS 3 NRs, we set two magnetic configurations, one is a FM arrangement with all magnetic moments in the same direction and the other is an AFM configuration with the magnetic moments of the two edges anti-parallel. From the results of the total energy calculation, we can easily find that the ground state of N-a-TiS 3 NRs are not all FM which is different from the previous theoretical results [31]. The magnetic ground state varies as the width changes. Figure 3a shows that the energy difference between the magnetic state and nonmagnetic state, one can easily see that when N is odd, the ground state is FM, meanwhile N is even, the ground state is AFM. This width-dependent magnetic ground state is different from other 2D material NRs like black phosphorus [23]. chain whose terminal atoms are Ti atom. The magnetic moment decreases from 0.144 μB (edge Ti atom) to 0.131 μB (center Ti atom). The distribution of magnetic moment also indicates the strong hybridization between Ti atoms. Figure 3f,g are the spin density of 10-a-TiS3 NR in AFM ground state. The magnetic moment is mainly located at edge S atoms and Ti atoms. Due to the strong hybridization between Ti atoms in the same TiS3 chain, magnetic moments turn to zero gradually from the terminal Ti to center one.  Our calculations also reveal that all N-a-TiS 3 NRs have metallic behavior with bands crossing the Fermi level. The electronic band dispersion of ground magnetic states of 9-a-TiS 3 NR and 10-a-TiS 3 NR are shown in Figure 3b,c, respectively. For FM ground state of 9-a-TiS 3 , there is a spin split near the Fermi level. The total magnetic moment of the unit cell is 0.86 µB which is consistent with the previous calculation [31]. Due to the AFM ground state, two spin components of 10-a-TiS 3 are exactly overlapped indicating a perfect AFM state. We also checked the effect of SOC, and we found that the SOC does not change the magnetic ground states of N-a-TiS3 NRs, due to the small SOC effect in light elements.
To get further insight into the spin polarization in a-TiS 3 NRs, we also calculated the spin density, which is the difference between spin-up and spin-down channels, and show them in Figure 3d-g. Figure 3d,e are the spin density of FM 9-a-TiS 3 NR. The magnetic moment is mainly located on edge S and edge Ti atoms; in addition, the magnetic moment is still located in Ti atoms in the TiS 3 chain whose terminal atoms are Ti atom. The magnetic moment decreases from 0.144 µB (edge Ti atom) to 0.131 µB (center Ti atom). The distribution of magnetic moment also indicates the strong hybridization between Ti atoms. Figure 3f,g are the spin density of 10-a-TiS 3 NR in AFM ground state. The magnetic moment 6 of 8 is mainly located at edge S atoms and Ti atoms. Due to the strong hybridization between Ti atoms in the same TiS 3 chain, magnetic moments turn to zero gradually from the terminal Ti to center one.
As is well known, the edge states usually are sensitive to strain, doping, and the external field. Here we focused on the effect of strain on the edge states. A uniaxial strain varying from −6% to 6% was applied to investigate the magnetic and electronic properties. The energy differences between FM and nonmagnetic/AFM states of 9-a-TiS 3 NR (10-a-TiS 3 NR) are shown in Figure 4a. We found that for both 9 and 10-a-TiS 3 NR, the compressive strains have a small effect on the magnetic ground state. Although the FM ground state is not changed by tensile strain, the electronic structure had a dramatic change. We calculated the band structure of 9-a-TiS 3 NR under the various strains, and found that when a tensile strain as large as 6%, FM 9-a-TiS 3 NR become a half metal. The band structure of 9-a-TiS 3 NR under 6% tensile strain is plotted in Figure 4b with the red lines denoting the spin up component and blue dashed lines denoting the spin down component. We can clearly see that the spin-up component opens an indirect band gap about 0.2 eV while three spin-down bands cross Fermi energy indicating metal property. The energy difference between AFM and FM show that there is an AFM-FM phase transition of 10-a-TiS 3 NR under a 4% tensile strain. We also plotted the band structure of 10-a-TiS 3 NR under the 6% tensile strain, a large spin-split around Fermi energy shows the FM property.
Materials 2019, 12, x FOR PEER REVIEW 6 of 8 As is well known, the edge states usually are sensitive to strain, doping, and the external field. Here we focused on the effect of strain on the edge states. A uniaxial strain varying from −6% to 6% was applied to investigate the magnetic and electronic properties. The energy differences between FM and nonmagnetic/AFM states of 9-a-TiS3 NR (10-a-TiS3 NR) are shown in Figure 4a. We found that for both 9 and 10-a-TiS3 NR, the compressive strains have a small effect on the magnetic ground state. Although the FM ground state is not changed by tensile strain, the electronic structure had a dramatic change. We calculated the band structure of 9-a-TiS3 NR under the various strains, and found that when a tensile strain as large as 6%, FM 9-a-TiS3 NR become a half metal. The band structure of 9-a-TiS3 NR under 6% tensile strain is plotted in Figure 4b with the red lines denoting the spin up component and blue dashed lines denoting the spin down component. We can clearly see that the spin-up component opens an indirect band gap about 0.2 eV while three spin-down bands cross Fermi energy indicating metal property. The energy difference between AFM and FM show that there is an AFM-FM phase transition of 10-a-TiS3 NR under a 4% tensile strain. We also plotted the band structure of 10-a-TiS3 NR under the 6% tensile strain, a large spin-split around Fermi energy shows the FM property.

Conclusions
In summary, we investigated the electronic and magnetic properties of TiS3 NRs and strain effect by first-principles calculations. Our results reveal that b-TiS3 NRs are non-magnetic semiconductors consistent with previous results. However, the magnetic ground states of N-a-TiS3 NRs are strongly dependent on the width of NRs. When N is an (even) odd number, N-a-TiS3 NRs are (anti-)ferromagnetic metals. The strain effect is also investigated. The compressive strain has little effect on the electronic and magnetic properties of 9(10)-a-TiS3 NRs. On the contrary, a 6% tensile strain will tune 9-a-TiS3 NR from a FM metal into a half metal while a 4% tensile strain cause a AFM-FM phase transition of 10-a-TiS3 NR. Our findings make the N-a-TiS3 NRs promising spintronic devices.

Conclusions
In summary, we investigated the electronic and magnetic properties of TiS 3 NRs and strain effect by first-principles calculations. Our results reveal that b-TiS 3 NRs are non-magnetic semiconductors consistent with previous results. However, the magnetic ground states of N-a-TiS 3 NRs are strongly dependent on the width of NRs. When N is an (even) odd number, N-a-TiS 3 NRs are (anti-)ferromagnetic metals. The strain effect is also investigated. The compressive strain has little effect on the electronic and magnetic properties of 9(10)-a-TiS 3 NRs. On the contrary, a 6% tensile strain will tune 9-a-TiS 3